Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
Einstein Metrics,
Harmonic Forms, &
Symplectic Four-Manifolds
Claude LeBrunStony Brook University
Sardegna, 2014/9/20
1
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
2
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
3
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
“. . . the greatest blunder of my life!”
— A. Einstein, to G. Gamow
4
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
5
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
As punishment . . .
6
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
λ called Einstein constant.
7
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
λ called Einstein constant.
Has same sign as the scalar curvature
s = rjj = Rijij.
8
Definition. A Riemannian metric h is said tobe Einstein if it has constant Ricci curvature —i.e.
r = λh
for some constant λ ∈ R.
λ called Einstein constant.
Has same sign as the scalar curvature
s = rjj = Rijij.
volg(Bε(p))
cnεn= 1− s ε2
6(n+2)+ O(ε4)
r......................................................................................... ε
.....................................................................................................................................................................................................................................................................................................................................................................................................................................
............................
.......................
..........................................................................................................................................................
....... .............
.......................................................................
9
Geometrization Problem:
10
Geometrization Problem:
Given a smooth compact manifold Mn, can one
11
Geometrization Problem:
Given a smooth compact manifold Mn, can onedecompose M in Einstein and collapsed pieces?
12
Geometrization Problem:
Given a smooth compact manifold Mn, can onedecompose M in Einstein and collapsed pieces?
When n = 3, answer is “Yes.”
13
Geometrization Problem:
Given a smooth compact manifold Mn, can onedecompose M in Einstein and collapsed pieces?
When n = 3, answer is “Yes.”
Proof by Ricci flow. Perelman et al.
14
Geometrization Problem:
Given a smooth compact manifold Mn, can onedecompose M in Einstein and collapsed pieces?
When n = 3, answer is “Yes.”
Proof by Ricci flow. Perelman et al.
Perhaps reasonable in other dimensions?
15
Recognition Problem:
16
Recognition Problem:
Suppose Mn admits Einstein metric h.
17
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
18
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
Can we recognize M by looking at h?
19
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
Can we recognize M by looking at h?
When n = 3, h has constant sectional curvature!
20
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
Can we recognize M by looking at h?
When n = 3, h has constant sectional curvature!
So M has universal cover S3, R3, H3. . .
21
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
Can we recognize M by looking at h?
When n = 3, h has constant sectional curvature!
So M has universal cover S3, R3, H3. . .
But when n ≥ 5, situation seems hopeless.
22
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
Can we recognize M by looking at h?
When n = 3, h has constant sectional curvature!
So M has universal cover S3, R3, H3. . .
But when n ≥ 5, situation seems hopeless.
{Einstein metrics on Sn}/∼ is highly disconnected.
23
Recognition Problem:
Suppose Mn admits Einstein metric h.
What, if anything, does h then tell us about M?
Can we recognize M by looking at h?
When n = 3, h has constant sectional curvature!
So M has universal cover S3, R3, H3. . .
But when n ≥ 5, situation seems hopeless.
{Einstein metrics on Sn}/∼ is highly disconnected.
When n ≥ 4, situation is more encouraging. . .
24
Moduli Spaces of Einstein metrics
25
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
26
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
Known to be connected for certain 4-manifolds:
27
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
Known to be connected for certain 4-manifolds:
M = T 4, K3, H4/Γ, CH2/Γ.
28
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
Known to be connected for certain 4-manifolds:
M = T 4, K3, H4/Γ, CH2/Γ.
Berger, Hitchin, Besson-Courtois-Gallot, L.
29
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
Known to be connected for certain 4-manifolds:
M = T 4, K3, H4/Γ, CH2/Γ.
Berger, Hitchin, Besson-Courtois-Gallot, L.
30
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
Known to be connected for certain 4-manifolds:
M = T 4, K3, H4/Γ, CH2/Γ.
Berger, Hitchin, Besson-Courtois-Gallot, L.
31
Moduli Spaces of Einstein metrics
E (M) = {Einstein h}/(Diffeos× R+)
Known to be connected for certain 4-manifolds:
M = T 4, K3, H4/Γ, CH2/Γ.
Berger, Hitchin, Besson-Courtois-Gallot, L.
32
Four Dimensions is Exceptional
33
Four Dimensions is Exceptional
When n = 4, Einstein metrics are genuinely non-trivial: not typically spaces of constant curvature.
34
Four Dimensions is Exceptional
When n = 4, Einstein metrics are genuinely non-trivial: not typically spaces of constant curvature.
There are beautiful and subtle global obstructionsto the existence of Einstein metrics on 4-manifolds.
35
Four Dimensions is Exceptional
When n = 4, Einstein metrics are genuinely non-trivial: not typically spaces of constant curvature.
There are beautiful and subtle global obstructionsto the existence of Einstein metrics on 4-manifolds.
But might allow for geometrization of 4-manifoldsby decomposition into Einstein and collapsed pieces.
36
Four Dimensions is Exceptional
When n = 4, Einstein metrics are genuinely non-trivial: not typically spaces of constant curvature.
There are beautiful and subtle global obstructionsto the existence of Einstein metrics on 4-manifolds.
But might allow for geometrization of 4-manifoldsby decomposition into Einstein and collapsed pieces.
One key question:
37
Four Dimensions is Exceptional
When n = 4, Einstein metrics are genuinely non-trivial: not typically spaces of constant curvature.
There are beautiful and subtle global obstructionsto the existence of Einstein metrics on 4-manifolds.
But might allow for geometrization of 4-manifoldsby decomposition into Einstein and collapsed pieces.
One key question:
Does enough rigidity really hold in dimension fourto make this a genuine geometrization?
38
Symplectic 4-manifolds:
39
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
40
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
41
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
42
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.
43
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.
Some Suggestive Questions. If (M4, ω) is asymplectic 4-manifold, when does M4 admit anEinstein metric h (unrelated to ω)? What if wealso require λ ≥ 0?
44
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.
Some Suggestive Questions. If (M4, ω) is asymplectic 4-manifold, when does M4 admit anEinstein metric h (unrelated to ω)? What if wealso require λ ≥ 0?
45
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.
Some Suggestive Questions. If (M4, ω) is asymplectic 4-manifold, when does M4 admit anEinstein metric h (unrelated to ω)? What if wealso require λ ≥ 0?
46
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.
Some Suggestive Questions. If (M4, ω) is asymplectic 4-manifold, when does M4 admit anEinstein metric h (unrelated to ω)? What if wealso require λ ≥ 0?
47
Symplectic 4-manifolds:
A laboratory for exploring Einstein metrics.
Kahler geometry is a rich source of examples.
If M admits a Kahler metric, it of course admits asymplectic form ω.
On such manifolds, Seiberg-Witten theory mimicsKahler geometry when treating non-Kahler metrics.
Some Suggestive Questions. If (M4, ω) is asymplectic 4-manifold, when does M4 admit anEinstein metric h (unrelated to ω)? What if wealso require λ ≥ 0?
48
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
49
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
50
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
51
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
52
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
53
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
54
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
55
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
56
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2
K3
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
57
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
58
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
59
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
60
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
61
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
62
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
63
Conventions:
CP2 = reverse oriented CP2.
64
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
...........................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
................... ...........................
................................................................................................................................................................................................................................................................................................................................................................................
.........................
..................................................
...
................................................................................. .....................................................
......................................................................................................................................
65
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
.....................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
......................................................................................................................................
................................................................. ..................................
................................
................................
................................
................................
66
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
.....................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
......................................................................................................................................
..................................
..................................
................................
................................
67
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
.....................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
......................................................................................................................................
................................................................. ..................................
................................
................................
................................
................................
68
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
...........................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................
................... ...........................
................................................................................................................................................................................................................................................................................................................................................................................
.........................
..................................................
...
................................................................................. .....................................................
......................................................................................................................................
69
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
.....................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
......................................................................................................................................
................................................................. ..................................
................................
................................
................................
................................
70
Conventions:
CP2 = reverse oriented CP2.
Connected sum #:
.....................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................
................
................................................................................. .....................................................
......................................................................................................................................
..................................
..................................
................................
................................
71
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
72
Theorem (L ’09). Suppose that M is a smoothcompact oriented 4-manifold which admits asymplectic structure ω. Then M also admits anEinstein metric h with λ ≥ 0 if and only if
Mdiff≈
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,
K3,
K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Del Pezzo surfaces,K3 surface, Enriques surface,Abelian surface, Hyper-elliptic surfaces.
73
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
74
Definitive list . . .
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
75
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
76
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
77
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
78
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
79
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
80
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M)= {Einstein metrics}/Diffeomorphisms
81
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) = {Einstein h}/(Diffeos×R+)
82
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) completely understood.
83
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) connected!
84
But we understand some cases better than others!
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) connected!
85
Above the line:
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) connected!
86
Above the line:
Know an Einstein metric on each manifold.
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) connected!
87
Above the line:
Moduli space E (M) 6= ∅. Is it connected?
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) connected!
88
Above the line:
Moduli space E (M) 6= ∅. But is it connected?
M ===
CP2#kCP2, 0 ≤ k ≤ 8,
S2 × S2,K3,K3/Z2,
T 4,
T 4/Z2, T4/Z3, T
4/Z4, T4/Z6,
T 4/(Z2 ⊕ Z2), T 4/(Z3 ⊕ Z3), or T 4/(Z2 ⊕ Z4).
Below the line:
Every Einstein metric is Ricci-flat Kahler.
Moduli space E (M) connected!
89
Objective of this lecture:
90
Objective of this lecture:
Describe modest recent progress on this issue.
91
Objective of this lecture:
Describe modest recent progress on this issue.
For M a Del Pezzo surface, will define explicit open
92
Objective of this lecture:
Describe modest recent progress on this issue.
For M a Del Pezzo surface, will define explicit open
U ⊂ {Riemannian metrics on M}
93
Objective of this lecture:
Describe modest recent progress on this issue.
For M a Del Pezzo surface, will define explicit open
U ⊂ {Riemannian metrics on M}
such that
94
Objective of this lecture:
Describe modest recent progress on this issue.
For M a Del Pezzo surface, will define explicit open
U ⊂ {Riemannian metrics on M}
such that
• Every known Einstein metric belongs to U ;
95
Objective of this lecture:
Describe modest recent progress on this issue.
For M a Del Pezzo surface, will define explicit open
U ⊂ {Riemannian metrics on M}
such that
• Every known Einstein metric belongs to U ;
• These form a connected family, mod diffeos; and
96
Objective of this lecture:
Describe modest recent progress on this issue.
For M a Del Pezzo surface, will define explicit open
U ⊂ {Riemannian metrics on M}
such that
• Every known Einstein metric belongs to U ;
• These form a connected family, mod diffeos; and
•No other Einstein metrics belong to U !
97
Formulation will depend on. . .
98
Special character of dimension 4:
99
Special character of dimension 4:
The Lie group SO(4) is not simple:
100
Special character of dimension 4:
The Lie group SO(4) is not simple:
so(4) ∼= so(3)⊕ so(3).
101
Special character of dimension 4:
The Lie group SO(4) is not simple:
so(4) ∼= so(3)⊕ so(3).
On oriented (M4, g), =⇒
102
Special character of dimension 4:
The Lie group SO(4) is not simple:
so(4) ∼= so(3)⊕ so(3).
On oriented (M4, g), =⇒Λ2 = Λ+ ⊕ Λ−
103
Special character of dimension 4:
The Lie group SO(4) is not simple:
so(4) ∼= so(3)⊕ so(3).
On oriented (M4, g), =⇒Λ2 = Λ+ ⊕ Λ−
where Λ± are (±1)-eigenspaces of
? : Λ2→ Λ2,
104
Special character of dimension 4:
The Lie group SO(4) is not simple:
so(4) ∼= so(3)⊕ so(3).
On oriented (M4, g), =⇒Λ2 = Λ+ ⊕ Λ−
where Λ± are (±1)-eigenspaces of
? : Λ2→ Λ2,
?2 = 1.
Λ+ self-dual 2-forms.Λ− anti-self-dual 2-forms.
105
Riemann curvature of g
R : Λ2→ Λ2
106
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
107
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
Λ+∗ Λ−∗
Λ+ W+ + s12 r
Λ− r W− + s12
108
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
Λ+∗ Λ−∗
Λ+ W+ + s12 r
Λ− r W− + s12
where
s = scalar curvature
r = trace-free Ricci curvature
W+ = self-dual Weyl curvature (conformally invariant)
W− = anti-self-dual Weyl curvature
109
Riemann curvature of g
R : Λ2→ Λ2
splits into 4 irreducible pieces:
Λ+∗ Λ−∗
Λ+ W+ + s12 r
Λ− r W− + s12
where
s = scalar curvature
r = trace-free Ricci curvature
W+ = self-dual Weyl curvature (conformally invariant)
W− = anti-self-dual Weyl curvature ′′
110
Smooth compactM4 has invariants b±(M), definedin terms of intersection pairing
111
Smooth compactM4 has invariants b±(M), definedin terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
112
Smooth compactM4 has invariants b±(M), definedin terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
113
Smooth compactM4 has invariants b±(M), definedin terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
114
Smooth compactM4 has invariants b±(M), definedin terms of intersection pairing
H2(M,R)×H2(M,R) −→ R
( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ
Diagonalize:
+1. . .
+1︸ ︷︷ ︸b+(M)
b−(M)
−1. . .
−1
.
115
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ? ϕ = 0}.
116
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ? ϕ = 0}.Since ? is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
117
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ? ϕ = 0}.Since ? is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
where
H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.
118
Hodge theory:
H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ? ϕ = 0}.Since ? is involution of RHS, =⇒
H2(M,R) = H+g ⊕H−g ,
where
H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms. Then
b±(M) = dimH±g .
119
H+g
H−g
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..............................................................................................................................................
..............................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
........................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................................
{a | a · a = 0} ⊂H2(M,R)
120
H+g
H−g
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..............................................................................................................................................
..............................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.................................................................. ...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
{a | a · a = 0} ⊂ H2(M,R)
121
H+g′
H−g′
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
.................................
.....................................................................................................
....................................................................................................................................................................................................
....................................................................................................................................................................................................
............................
............................
............................
......
............................
............................
............................
............................
............................
......
...........................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................
.......................................................................................................................................
{a | a · a = 0} ⊂ H2(M,R)
122
H+g
H−g
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..............................................................................................................................................
..............................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.................................................................. ...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
b±(M) = dimH±g .
123
H+g′
H−g′
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................
..................................
.................................
.....................................................................................................
....................................................................................................................................................................................................
....................................................................................................................................................................................................
............................
............................
............................
......
............................
............................
............................
............................
............................
......
...........................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................
.......................................................................................................................................
b±(M) = dimH±g .
124
H+g
H−g
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..............................................................................................................................................
..............................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.................................................................. ...................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
b±(M) = dimH±g .
125
H+g
H−g
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
..............................................................................................................................................
..............................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
........................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................................
b±(M) = dimH±g .
126
More Technical Question. When does a com-pact complex surface (M4, J) admit an Einsteinmetric h which is Hermitian, in the sense that
127
More Technical Question. When does a com-pact complex surface (M4, J) admit an Einsteinmetric h which is Hermitian, in the sense that
h(·, ·) = h(J ·, J ·)?
128
More Technical Question. When does a com-pact complex surface (M4, J) admit an Einsteinmetric h which is Hermitian, in the sense that
h(·, ·) = h(J ·, J ·)?
Kahler if the 2-form
ω = h(J ·, ·)is closed:
dω = 0.
But we do not assume this!
129
More Technical Question. When does a com-pact complex surface (M4, J) admit an Einsteinmetric h which is Hermitian, in the sense that
h(·, ·) = h(J ·, J ·)?
130
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
131
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
132
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
133
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Aubin, Yau, Siu, Tian . . . Kahler case.
134
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Aubin, Yau, Siu, Tian . . . Kahler case.
Chen-L-Weber (2008), L (2012): non-Kahler case.
135
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Aubin, Yau, Siu, Tian . . . Kahler case.
Chen-L-Weber (2008), L (2012): non-Kahler case.
Only two metrics arise in non-Kahler case!
136
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Key Point. Metrics are actually conformally Kahler.
137
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Key Point. Metrics are actually conformally Kahler.
138
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Key Point. Metrics are actually conformally Kahler.
h = s−2g
139
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Key Point. Metrics are actually conformally Kahler.
Strictly 4-dimensional phenomenon!
140
Theorem. A compact complex surface (M4, J)admits an Einstein metric h which is Hermitianwith respect to J ⇐⇒ c1(M4, J) “has a sign.”
More precisely, ∃ such h with Einstein constantλ ⇐⇒ there is a Kahler form ω such that
c1(M4, J) = λ[ω].
Moreover, this metric is unique, up to isometry,if λ 6= 0.
Key Point. Metrics are actually conformally Kahler.
Strictly 4-dimensional phenomenon!
“Riemannian Goldberg-Sachs Theorem.”
141
So, which complex surfaces admit
142
So, which complex surfaces admit
Einstein Hermitian metrics with λ > 0?
143
Del Pezzo surfaces:
144
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].
145
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
146
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
147
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
148
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
149
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
.......
.......
.......
.......
.......
.......
.................................................................................................................................................................
....................
......................
.........................
................................
................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
......................................................................................................................................................................................................................................................................................
CP2
•
••
....................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................
.......................................................................................................................
..................................................................................................................................................................................................................................................
.....................................................................................................
.....................................................................................................
150
Blowing up:
151
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
152
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
153
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
154
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
M ≈ N#CP2
in which added CP1 has normal bundle O(−1).
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
155
156
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
M ≈ N#CP2
in which added CP1 has normal bundle O(−1).
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
157
158
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
M ≈ N#CP2
in which added CP1 has normal bundle O(−1).
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
159
160
Blowing up:
If N is a complex surface, may replace p ∈ Nwith CP1 to obtain blow-up
M ≈ N#CP2
in which added CP1 has normal bundle O(−1).
rN
M............................................................................................................... ..........
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................
.....................................................
......................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................
.............................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.........................................
161
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
.......
.......
.......
.......
.......
.......
.................................................................................................................................................................
....................
......................
.........................
................................
................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
......................................................................................................................................................................................................................................................................................
CP2
•
••
....................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................
.......................................................................................................................
..................................................................................................................................................................................................................................................
.....................................................................................................
.....................................................................................................
162
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
.......
.......
.......
.......
.......
.......
.................................................................................................................................................................
....................
......................
.........................
................................
................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
......................................................................................................................................................................................................................................................................................
CP2
•
••
....................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................
.......................................................................................................................
..................................................................................................................................................................................................................................................
.....................................................................................................
.....................................................................................................
No 3 on a line, no 5 on conic, no 8 on nodal cubic.
163
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
.......
.......
.......
.......
.......
.......
.................................................................................................................................................................
....................
......................
.........................
................................
................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
......................................................................................................................................................................................................................................................................................
CP2
••
••
•
• .........................................
........................
...................................
................................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................
.......................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................
No 3 on a line, no 6 on conic, no 8 on nodal cubic.
164
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
.......
.......
.......
.......
.......
.......
.................................................................................................................................................................
....................
......................
.........................
................................
................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................
......................................................................................................................................................................................................................................................................................
CP2
••
••
••
•
• ....................................................................................
...................................................................................................................................................................................................................................................................................
.....................
...................
........................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................
No 3 on a line, no 6 on conic, no 8 on nodal cubic.
165
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
166
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
For each topological type:
167
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
For each topological type:
Moduli space of such (M4, J) is connected.
168
Del Pezzo surfaces:
(M4, J) for which c1 is a Kahler class [ω].Shorthand: “c1 > 0.”
Blow-up of CP2 at k distinct points, 0 ≤ k ≤ 8,in general position, or CP1 × CP1.
For each topological type:
Moduli space of such (M4, J) is connected.
Just a point if b2(M) ≤ 5.
169
Every del Pezzo surface has b+ = 1. ⇐⇒
170
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
171
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
172
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
173
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
174
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
175
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
176
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
177
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
178
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
179
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
180
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
• open condition;
181
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
• open condition;
• conformally invariant; and
182
Every del Pezzo surface has b+ = 1. ⇐⇒
Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:
dω = 0, ?ω = ω.
Such a form defines a symplectic structure, exceptat points where ω = 0.
Definition. Let M be smooth 4-manifold withb+(M) = 1. We will say that a Riemannianmetric h on M is of symplectic type if associatedSD harmonic ω is nowhere zero.
• open condition;
• conformally invariant; and
• holds in Kahler case.
183
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
184
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
185
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
186
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
187
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
188
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
189
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Notice that
190
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Notice that
• =⇒ ω is symplectic form.
191
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Notice that
• =⇒ ω is symplectic form.
• =⇒ W+ is everywhere non-zero.
192
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
193
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
However,
194
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
However,
• recall that W+ is trace-free; so
195
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
However,
• recall that W+ is trace-free; so
• wherever W+ non-zero, has positive directions.
196
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
197
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Thus, we are really just demanding that
198
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Thus, we are really just demanding that
• ω 6= 0 everywhere;
199
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Thus, we are really just demanding that
• ω 6= 0 everywhere;
•W+ 6= everywhere; and
200
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
Thus, we are really just demanding that
• ω 6= 0 everywhere;
•W+ 6= everywhere; and
•W+ and ω are everywhere roughly aligned.
201
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
202
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
• open condition;
203
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
• open condition;
• conformally invariant; and
204
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
• open condition;
• conformally invariant; and
• holds for (almost) Kahler metrics with s > 0.
205
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
• open condition;
• conformally invariant; and
• holds for (almost) Kahler metrics with s > 0.
Kahler =⇒ W+ =
− s12− s
12s6
206
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
• open condition;
• conformally invariant; and
• holds for (almost) Kahler metrics with s > 0.
207
Definition. Let M be smooth compact 4-manifoldwith b+(M) = 1. We will say that a Riemannianmetric h is of positive symplectic type if Weylcurvature and SD harmonic form ω satisfy
W+(ω, ω) > 0
at every point of M .
• open condition;
• conformally invariant; and
• holds for (almost) Kahler metrics with s > 0.
208
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
209
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
210
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
211
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
212
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
213
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
In other words, h is known, and is either
214
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
In other words, h is known, and is either
•Kahler-Einstein, with λ > 0; or
215
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
In other words, h is known, and is either
•Kahler-Einstein, with λ > 0; or
• Page metric on CP2#CP2; or
216
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
In other words, h is known, and is either
•Kahler-Einstein, with λ > 0; or
• Page metric on CP2#CP2; or
• CLW metric on CP2#2CP2.
217
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
In other words, h is known, and is either
•Kahler-Einstein, with λ > 0; or
• Page metric on CP2#CP2; or
• CLW metric on CP2#2CP2.
Conversely, all these are of positive symplectic type.
218
For M4 a Del Pezzo surface, set
219
For M4 a Del Pezzo surface, set
considered as smooth compact oriented 4-manifold,
220
For M4 a Del Pezzo surface, set
221
For M4 a Del Pezzo surface, set
222
For M4 a Del Pezzo surface, set
E (M) = {Einstein h on M}/(Diffeos× R+)
223
For M4 a Del Pezzo surface, set
E (M) = {Einstein h on M}/(Diffeos× R+)
E +ω (M) = {Einstein h with W+(ω, ω) > 0}/∼
224
For M4 a Del Pezzo surface, set
E (M) = {Einstein h on M}/(Diffeos× R+)
E +ω (M) = {Einstein h with W+(ω, ω) > 0}/∼
Corollary. E +ω (M) is connected. Moreover, if
b2(M) ≤ 5, then E +ω (M) = {point}.
225
For M4 a Del Pezzo surface, set
E (M) = {Einstein h on M}/(Diffeos× R+)
E +ω (M) = {Einstein h with W+(ω, ω) > 0}/∼
Corollary. E +ω (M) is connected. Moreover, if
b2(M) ≤ 5, then E +ω (M) = {point}.
226
For M4 a Del Pezzo surface, set
E (M) = {Einstein h on M}/(Diffeos× R+)
E +ω (M) = {Einstein h with W+(ω, ω) > 0}/∼
Corollary. E +ω (M) is connected. Moreover, if
b2(M) ≤ 5, then E +ω (M) = {point}.
227
For M4 a Del Pezzo surface, set
E (M) = {Einstein h on M}/(Diffeos× R+)
E +ω (M) = {Einstein h with W+(ω, ω) > 0}/∼
Corollary. E +ω (M) is connected. Moreover, if
b2(M) ≤ 5, then E +ω (M) = {point}.
Corollary.E +ω (M) is exactly one connected com-
ponent of E (M).
228
Method of Proof.
229
Key Observation:
230
Key Observation:
By second Bianchi identity,
231
Key Observation:
By second Bianchi identity,
h Einstein =⇒ δW+ = (δW )+ = 0.
232
Key Observation:
By second Bianchi identity,
h Einstein =⇒ δW+ = (δW )+ = 0.
(δW )bcd := −∇aW abcd = −∇[crd]b +
1
6hb[c∇d]s
233
Key Observation:
By second Bianchi identity,
h Einstein =⇒ δW+ = (δW )+ = 0.
(δW )bcd := −∇aW abcd = −∇[crd]b +
1
6hb[c∇d]s
Our strategy:
234
Key Observation:
By second Bianchi identity,
h Einstein =⇒ δW+ = (δW )+ = 0.
(δW )bcd := −∇aW abcd = −∇[crd]b +
1
6hb[c∇d]s
Our strategy:
study weaker equation
235
Key Observation:
By second Bianchi identity,
h Einstein =⇒ δW+ = (δW )+ = 0.
(δW )bcd := −∇aW abcd = −∇[crd]b +
1
6hb[c∇d]s
Our strategy:
study weaker equation
δW+ = 0
236
Key Observation:
By second Bianchi identity,
h Einstein =⇒ δW+ = (δW )+ = 0.
(δW )bcd := −∇aW abcd = −∇[crd]b +
1
6hb[c∇d]s
Our strategy:
study weaker equation
δW+ = 0
as proxy for Einstein equation.
237
Now suppose that ω 6= 0 everywhere.
238
Now suppose that ω 6= 0 everywhere.
Rescale h to obtain g with |ω| ≡√
2:
239
Now suppose that ω 6= 0 everywhere.
Rescale h to obtain g with |ω| ≡√
2:
g =1√2|ω|h.
240
Now suppose that ω 6= 0 everywhere.
Rescale h to obtain g with |ω| ≡√
2:
g =1√2|ω|h.
This g is almost-Kahler:
241
Now suppose that ω 6= 0 everywhere.
Rescale h to obtain g with |ω| ≡√
2:
g =1√2|ω|h.
This g is almost-Kahler:
related to symplectic form ω by
242
Now suppose that ω 6= 0 everywhere.
Rescale h to obtain g with |ω| ≡√
2:
g =1√2|ω|h.
This g is almost-Kahler:
related to symplectic form ω by
g = ω(·, J ·)
243
Now suppose that ω 6= 0 everywhere.
Rescale h to obtain g with |ω| ≡√
2:
g =1√2|ω|h.
This g is almost-Kahler:
related to symplectic form ω by
g = ω(·, J ·)
for some g-preserving almost-complex structure J .
244
Weitzenbock for harmonic self-dual 2-form ω:
245
Weitzenbock for harmonic self-dual 2-form ω:
0 = ∇∗∇ω − 2W+(ω, ·) +s
3ω
246
Weitzenbock for harmonic self-dual 2-form ω:
0 = ∇∗∇ω − 2W+(ω, ·) +s
3ω
Taking inner product with ω:
247
Weitzenbock for harmonic self-dual 2-form ω:
0 = ∇∗∇ω − 2W+(ω, ·) +s
3ω
Taking inner product with ω:
0 =1
2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +
s
3|ω|2
248
Weitzenbock for harmonic self-dual 2-form ω:
0 = ∇∗∇ω − 2W+(ω, ·) +s
3ω
Taking inner product with ω:
0 =1
2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +
s
3|ω|2
In almost-Kahler case, |ω|2 ≡ 2, so
249
Weitzenbock for harmonic self-dual 2-form ω:
0 = ∇∗∇ω − 2W+(ω, ·) +s
3ω
Taking inner product with ω:
0 =1
2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +
s
3|ω|2
In almost-Kahler case, |ω|2 ≡ 2, so
1
2|∇ω|2 = W+(ω, ω)− s
3
250
Weitzenbock for harmonic self-dual 2-form ω:
0 = ∇∗∇ω − 2W+(ω, ·) +s
3ω
Taking inner product with ω:
0 =1
2∆|ω|2 + |∇ω|2 − 2W+(ω, ω) +
s
3|ω|2
In almost-Kahler case, |ω|2 ≡ 2, so
1
2|∇ω|2 = W+(ω, ω)− s
3
W+(ω, ω) ≥ s
3
251
Equation δW+ = 0? not conformally invariant!
252
Equation δW+ = 0 not conformally invariant!
253
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
254
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
δW+ = 0
255
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
δW+ = 0
then g instead satisfies
256
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
δW+ = 0
then g instead satisfies
δ(fW+) = 0
257
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
δW+ = 0
then g instead satisfies
δ(fW+) = 0
which in turn implies the Weitzenbock formula
258
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
δW+ = 0
then g instead satisfies
δ(fW+) = 0
which in turn implies the Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
259
Equation δW+ = 0 not conformally invariant!
If h = f2g satisfies
δW+ = 0
then g instead satisfies
δ(fW+) = 0
which in turn implies the Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
for fW+ ∈ End(Λ+).
260
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
261
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω⊗ω and integrate by parts, using identity
262
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω⊗ω and integrate by parts, using identity
263
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
264
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
265
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
266
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
0 =
∫M
(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2
)f dµ,
267
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
0 =
∫M
(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2
)f dµ,
where W+(ω)⊥ = projection of W+(ω, ·) to ω⊥.
268
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
0 =
∫M
(−sW+(ω, ω) + 8|W+|2 − 4|W+(ω)⊥|2
)f dµ
269
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
0 ≥∫M
(−sW+(ω, ω) + 3
[W+(ω, ω)
]2 )f dµ
270
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
0 ≥ 3
∫MW+(ω, ω)
(W+(ω, ω)− s
3
)f dµ
271
Now take inner product of Weitzenbock formula
0 = ∇∗∇(fW+) +s
2fW+ − 6fW+ ◦W+ + 2f |W+|2I
with 2ω ⊗ ω and integrate by parts, using identity
〈W+,∇∗∇(ω⊗ω)〉 = [W+(ω, ω)]2+4|W+(ω)|2−sW+(ω, ω) .
This yields
0 ≥ 3
∫MW+(ω, ω)
(1
2|∇ω|2
)f dµ
272
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
273
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
274
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
275
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
276
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
277
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
278
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
279
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
280
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
281
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
282
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
283
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
284
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
285
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
Kahler =⇒ W+ =
− s12− s
12s6
286
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
287
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
Conversely, any Kahler (M4, g, ω) with s > 0 sat-isfies W+(ω, ω) > 0, and f = c/s solves
288
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
Conversely, any Kahler (M4, g, ω) with s > 0 sat-isfies W+(ω, ω) > 0, and f = c/s solves
289
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
Conversely, any Kahler (M4, g, ω) with s > 0 sat-isfies W+(ω, ω) > 0, and f = c/s solves
∇(fW+) = 0 =⇒ δ(fW+) = 0.
290
Proposition. If compact almost-Kahler (M4, g, ω)satisfies δ(fW+) = 0 for some f > 0, then
0 ≥∫MW+(ω, ω)|∇ω|2f dµ
Corollary. Let (M4, g, ω) be a compact almost-Kahler manifold with W+(ω, ω) > 0. If
δ(fW+) = 0
for some f > 0, then g is a Kahler metric withscalar curvature s > 0. Moreover, f = c/s forsome constant c > 0.
Conversely, any Kahler (M4, g, ω) with s > 0 sat-isfies W+(ω, ω) > 0, and f = c/s solves
∇(fW+) = 0 =⇒ δ(fW+) = 0.
291
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
292
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
293
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
294
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
295
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
296
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
297
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
298
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
299
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
300
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
301
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
302
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
303
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
304
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
305
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
306
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
307
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
308
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
309
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
Remark. If such metrics exist, b+(M) = 1.
310
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
311
Theorem. Let (M,h) be a compact oriented Rie-mannian 4-manifold with δW+ = 0. If
W+(ω, ω) > 0
for some self-dual harmonic 2-form ω, then
h = s−2g
for a unique Kahler metric g of scalar curvatures > 0.
Conversely, for any Kahler metric g of positivescalar curvature, the conformally related metrich = s−2g satisfies δW+ = 0 and W+(ω, ω) > 0.
Theorem A follows by restricting to Einstein case.
312
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
313
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
Corollary. E +ω (M) is connected. Moreover, if
b2(M) ≤ 5, then E +ω (M) = {point}.
314
Theorem A. Let (M,h) be a smooth compact4-dimensional Einstein manifold. If h is of posi-tive symplectic type, then it’s a conformally Kahler,Einstein metric on a Del Pezzo surface (M,J).
Corollary. E +ω (M) is connected. Moreover, if
b2(M) ≤ 5, then E +ω (M) = {point}.
Corollary.E +ω (M) is exactly one connected com-
ponent of E (M).
315
Application to Almost-Kahler Geometry:
316
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
317
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
318
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
319
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
320
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
321
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
322
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
323
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
324
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
In particular, gives a new proof of the following:
325
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
In particular, gives a new proof of the following:
Corollary (Sekigawa). Every compact almost-KahlerEinstein 4-manifold with non-negative Einsteinconstant is Kahler-Einstein.
326
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
In particular, gives a new proof of the following:
Corollary (Sekigawa). Every compact almost-KahlerEinstein 4-manifold with non-negative Einsteinconstant is Kahler-Einstein.
(Special case of “Goldberg conjecture.”)
327
Theorem. Let (M, g, ω) be a compact almost-Kahler 4-manifold with
s ≥ 0, δW+ = 0.
Then (M, g, ω) is a constant-scalar-curvature Kahler(“cscK”) manifold.
In particular, gives a new proof of the following:
Corollary (Sekigawa). Every compact almost-KahlerEinstein 4-manifold with non-negative Einsteinconstant is Kahler-Einstein.
Helped motivate the discovery of Theorem A. . .
328
Tanti auguri, Stefano!
329
Tanti auguri, Stefano!
E grazie agli organizzatori
330
Tanti auguri, Stefano!
E grazie agli organizzatori
per un convegno cosı bello!
331