Special holonomy, rational formality and fibrations€¦ · Manuel Amann Special holonomy, rational formality and brations. Special holonomy Formality ... Compact K ahler manifolds

Special holonomy, rational formality andfibrations

Manuel Amann1

Karlsruhe Institute of Technology

September 2012

1joint with Vitali Kapovitch

Special holonomyFormality

Fibrations and the main result

Special holonomy, rational formality andfibrations

Manuel Amann1

Karlsruhe Institute of Technology

September 2012

1joint with Vitali KapovitchManuel Amann Special holonomy, rational formality and fibrations



Parallel transport

Manuel Amann Special holonomy, rational formality and fibrations



The holonomy group

Let (M, g) be a Riemannian manifold.

Definition

The group

Holx(M) = {Pγ | Pγ is parallel transport along γ, a closed loop at x}⊆GL(TxM)

is the holonomy group of M (at x).




Special holonomy

Theorem (Berger)

The holonomy group of a simply-connected irreducible non-symmetricRiemannian manifold is one of the following.

Hol(M, g) dimM

SO(n) n genericU(n) 2n KahlerSU(n) 2n Calabi–YauSp(n)Sp(1) 4n Quaternion KahlerSp(n) 4n hyperKahlerG2 7 G2

Spin(7) 8 Spin(7)




Positive Quaternion Kahler Manifolds

Definition (Quaternion Kahler Manifold)

(Suppose n > 1.) A Quaternion Kahler Manifold is a connected orientedRiemannian manifold (M4n, g) with holonomy Hol(M, g)⊆Sp(n)Sp(1).

Quaternion Kahler Manifolds are Einstein, i.e. Ricg = λ · g for someλ ∈ R. In particular, their scalar curvature is constant.

Definition (Positive Quaternion Kahler Manifold)

A Positive Quaternion Kahler Manifold is a Quaternion Kahler Manifoldwith complete metric and positive scalar curvature.






















The basic conjecture

Positive Quaternion Kahler Manifolds are not Kahlerian in general.

The structure group reduces to Sp(n) if and only if the scalarcurvature vanishes.

The only known examples of Positive Quaternion Kahler Manifoldsare symmetric.

In each dimension there are only finitely many Positive QuaternionKahler manifolds (LeBrun, Salamon).

Conjecture (LeBrun, Salamon)

Each Positive Quaternion Kahler Manifold is symmetric.












































. . . with the conjecture being widely open in general. . .

Question

Which properties do Positive Quaternion Kahler Manifolds share withsymmetric spaces?

Let us address this question from a viewpoint completely new to thesubject: Rational Homotopy Theory




. . . with the conjecture being widely open in general. . .

Question

Which properties do Positive Quaternion Kahler Manifolds share withsymmetric spaces?

Let us address this question from a viewpoint completely new to thesubject: Rational Homotopy Theory




Cohomology vs. Homotopy

H∗(M,Z) π∗(M)





H∗(M,Z)gg

hur

H∗(M,Z)ww

univ. Koeff., PD

77

oo???

// π∗(M)





H∗(M,Z)gg

hur

H∗(M,Z)ww

univ. Koeff., PD

77

⊗R��

oo???

// π∗(M)

⊗R��

H∗(M,R) π∗(M)⊗ R





H∗(M,Z)gg

hur

H∗(M,Z)ww

univ. Koeff., PD

77

⊗R��

oo???

// π∗(M)

⊗R��

H∗(M,R) π∗(M)⊗ R

H(ADR(M),d)





H∗(M,Z)gg

hur

H∗(M,Z)ww

univ. Koeff., PD

77

⊗R��

oo???

// π∗(M)

⊗R��

H∗(M,R) π∗(M)⊗ R

H(ADR(M),d) (ADR(M),d)

RHT

OO





H∗(M,Z)gg

hur

H∗(M,Z)ww

univ. Koeff., PD

77

⊗R��

oo???

// π∗(M)

⊗R��

H∗(M,R) π∗(M)⊗ R

H(ADR(M),d) oo'

M formal?// (ADR(M),d)

RHT

OO




Formality

Definition

A manifold is formal if its rational homotopy type is a formalconsequence of its rational cohomology algebra.

That is in particular, we can compute its rational homotopy groupsfrom its rational cohomology algebra.

All its Massey products vanish.




Formality

Definition







Formality

Definition







Formality and special holonomy

Let us revisit Berger’s theorem.

Symmetric spaces are formal (Cartan).

A finite product is formal if and only if so are the factors.

Compact Kahler manifolds are formal (Deligne, Griffiths, Morgan,Sullivan).

Hol(M, g) formal?

SO(n) no “special” holonomySU(n)⊆U(n) yesSp(n)⊆U(2n) yesSp(n)Sp(1)G2,Spin(7) formal if elliptic, open in general









Hol(M, g) formal?










Hol(M, g) formal?










Hol(M, g) formal?










Hol(M, g) formal?






Conjecture

A simply-connected compact Riemannian manifold of special holonomy isa formal space.









Hol(M, g) formal?










Hol(M, g) formal?

SO(n) no “special” holonomySU(n)⊆U(n) yesSp(n)⊆U(2n) yesSp(n)Sp(1) YES if PQKG2,Spin(7) formal if elliptic, open in general




Formality of Positive Quaternion Kahler Manifolds

Theorem

A Positive Quaternion Kahler Manifold is a formal space.

Proof.

We use the twistor fibration

S2 ↪→ Z →M

over a Positive Quaternion Kahler Manifold. The space Z is a compactKahler Manifold and formal, consequently.




Formality of Positive Quaternion Kahler Manifolds

Theorem

A Positive Quaternion Kahler Manifold is a formal space.

Proof.

We use the twistor fibration

S2 ↪→ Z →M

over a Positive Quaternion Kahler Manifold. The space Z is a compactKahler Manifold and formal, consequently.




Formality in fibrations

Clearly, we need a topological result relating formality properties of thetotal and the base space.

Theorem

Let

F ↪→ Ef−→ B

be a fibration of simply-connected topological spaces of finite type.Suppose that F is elliptic, formal and satisfies the Halperin conjecture.Then E is formal if and only if B is formal.Moreover, if B and E are formal, then the map f is formal.

The preceding geometric result is a simple corollary, since S2 satisfies theHalperin conjecture easily.






Theorem

Let

F ↪→ Ef−→ B








Theorem

Let

F ↪→ Ef−→ B






Remarks

Let me explain the theorem:

A simply-connected space X is rationally elliptic if π∗(X)⊗Q isfinite dimensional.

A simply-connected elliptic space F of positive Euler-characteristicsatisfies satisfies the Halperin conjecture if

F ↪→ E −→ B

is a fibration of simply-connected spaces, then

H∗(E;Q) = H∗(B;Q)⊗H∗(F ;Q)

as modules.




Remarks




F ↪→ E −→ B


H∗(E;Q) = H∗(B;Q)⊗H∗(F ;Q)

as modules.




Remarks




F ↪→ E −→ B


H∗(E;Q) = H∗(B;Q)⊗H∗(F ;Q)

as modules.




Remarks (continued)

As a generalisation, we say that a space X (not necessarily withχ(X) > 0) satisfies the Halperin conjecture if its rationalcohomology algebra does not admit any non-trivial derivations ofnegative degree.

With this definition, there are spaces satisfying the Halperinconjecture, which do not have positive Euler characteristic.

The Halperin conjecture was verified on several examples like

Hard-Lefschetz manifolds,homogeneous spaces G/H (with rkG = rkH),the “generic case” (of an elliptic space with positive Eulercharacteristic) and many more. (Elliptic spaces X with χ(X) > 0 areformal.)

There are counter-examples to the theorem if one drops “elliptic”.




Remarks (continued)









Remarks (continued)









Remarks (continued)









Remarks (continued)









Remarks (continued)









Remarks (continued)









Remarks (continued)









. . .

What about the other geometries, i.e. G2 and Spin(7)?

What about geometric formality?

What about positively curved manifolds?

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Thank you very much!

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Special holonomy, rational formality and fibrations€¦ · Manuel Amann Special holonomy, rational formality and brations. Special holonomy Formality ... Compact K ahler manifolds