Cohomology Of Holomorphic Pullback Line Bundles On Smooth ... · Motivation From Physics The Hypersurface Case The Codimension Two Case Summary And Future Work Cohomology Of Holomorphic

Motivation From PhysicsThe Hypersurface Case

The Codimension Two CaseSummary And Future Work

Cohomology Of Holomorphic Pullback LineBundles On Smooth And Compact Normal Toric

Varieties

Martin Bies

Feb 11, 2014

Martin Bies Cohomology Of Holomorphic Pullback LIne Bundles



Localised Zero Modes In Type IIB CompactificationSimplification: Towards Toric Varieties

Section 1

Motivation From Physics

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U(1) Charged Localised Zero Modes

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Intersecting D7-Branes

D7 brane Ba withU (1)a gauge theory La

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D7 brane Bb withU (1)b gauge theory Lb

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C

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CGauge enhancement on Cdescribed by La|C ⊗ Lb|C

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CGauge enhancement on Cdescribed by La|C ⊗ Lb|C

Consequence hep-th/0403166

Spectrum of masslesszero modes at C

H0 (C, La|C ⊗ Lb|C)H1 (C, La|C ⊗ Lb|C)

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U(1) Charged Localised Zero Modes - Simplified Setup

Ba with La

Bb with Lb

C

CY X3

XΣ Simplifying Assumptions

XΣ a smooth and compactnormal toric variety

C, Ba, Bb, X3 ⊂ XΣ

submanifolds

∃L ∈ Pic (XΣ) s.t.L|C = La|C ⊗ Lb|C

New Question

How to compute H i (C , L|C )?

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Ba with La

Bb with Lb

C

CY X3

XΣ

Simplifying Assumptions



submanifolds


New Question


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Ba with La

Bb with Lb

C

CY X3




submanifolds


New Question


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Ba with La

Bb with Lb

C

CY X3




submanifolds


New Question


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Ba with La

Bb with Lb

C

CY X3




submanifolds


New Question


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Ba with La

Bb with Lb

C

CY X3




submanifolds


New Question


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What Is A Toric Variety? Book by D. Cox, J. Little, H. Schenk ’Toric Varieties’

Toric Varieties Via Homogenisation

Every smooth and compact normal toric variety XΣ is given by

XΣ∼= (Cr − Z ) / (C∗)a

Example: Complex Projective Space

CPn ≡(Cn+1 − {0}

)/C∗

Why Smooth And Compact Normal Toric Varieties?

Let XΣ∼= (Cr − Z ) / (C∗)a. Then it holds

Pic (XΣ) ∼= Za.

H i (XΣ,OXΣ(v)) are finite dimensional vector spaces.

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XΣ∼= (Cr − Z ) / (C∗)a


CPn ≡(Cn+1 − {0}

)/C∗



Pic (XΣ) ∼= Za.


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XΣ∼= (Cr − Z ) / (C∗)a


CPn ≡(Cn+1 − {0}

)/C∗



Pic (XΣ) ∼= Za.


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Example: Computation Of Ambient Space Cohomologies

Ingredients

ambient space CP3

L = OCP3 (1)

cohomCalg hep-th/1003.5217, hep-th/1010.3717, math.AG/1006.0780, hep-th/1006.2392

I implemented a function into Mathematica which computes abasis of cohomology based on cohomCalg.

Result from Mathematica

H0(CP3,OCP3 (1)

)= {α1x1 + α2x2 + α3x3 , αi ∈ C} ∼= C3

H i(CP3,OCP3 (1)

)= 0 for i ≥ 1

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Ingredients

ambient space CP3

L = OCP3 (1)




H0(CP3,OCP3 (1)

)= {α1x1 + α2x2 + α3x3 , αi ∈ C} ∼= C3

H i(CP3,OCP3 (1)

)= 0 for i ≥ 1

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Ingredients

ambient space CP3

L = OCP3 (1)




H0(CP3,OCP3 (1)

)= {α1x1 + α2x2 + α3x3 , αi ∈ C} ∼= C3

H i(CP3,OCP3 (1)

)= 0 for i ≥ 1

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Ingredients

ambient space CP3

L = OCP3 (1)




H0(CP3,OCP3 (1)

)= {α1x1 + α2x2 + α3x3 , αi ∈ C} ∼= C3

H i(CP3,OCP3 (1)

)= 0 for i ≥ 1

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Questions so far?

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The Principle Answer Is . . .The Practical Answer

Section 2

The Hypersurface Case

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Task

XΣ

CY X3

Ingredients

Toric variety XΣ

L = OXΣ(D)

s̃1 ∈ H0 (XΣ,OXΣ(S1)) s.t.

X3 = {p ∈ XΣ , s̃1 (p) = 0}

Goal

Compute H i(X3, L|X3

).

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Task

XΣ

CY X3

Ingredients

Toric variety XΣ

L = OXΣ(D)

s̃1 ∈ H0 (XΣ,OXΣ(S1)) s.t.

X3 = {p ∈ XΣ , s̃1 (p) = 0}

Goal


).

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Task

XΣ

CY X3

Ingredients

Toric variety XΣ

L = OXΣ(D)

s̃1 ∈ H0 (XΣ,OXΣ(S1)) s.t.

X3 = {p ∈ XΣ , s̃1 (p) = 0}

Goal


).

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Task

XΣ

CY X3

Ingredients

Toric variety XΣ

L = OXΣ(D)

s̃1 ∈ H0 (XΣ,OXΣ(S1)) s.t.

X3 = {p ∈ XΣ , s̃1 (p) = 0}

Goal


).

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Hypersurface Case

Theorem To the proof

The following sequence is sheaf exact

0→ OXΣ(D − S1)︸︷︷︸=L′

⊗s̃1→ OXΣ(D)︸︷︷︸

=L

r→ OXΣ(D)|X3

→ 0

Consequence: There is a long exact cohomology sequence

0 H0(XΣ,L′) H0(XΣ,L) H0(X3, L|X3)

H1(XΣ,L′) H1(XΣ,L) H1(X3, L|X3)

H2(XΣ,L′) H2(XΣ,L) H2(X3, L|X3)

α0 β0

δ0

α1

α2

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Hypersurface Case

Theorem To the proof


0→ OXΣ(D − S1)︸︷︷︸=L′

⊗s̃1→ OXΣ(D)︸︷︷︸

=L

r→ OXΣ(D)|X3

→ 0

Consequence: There is a long exact cohomology sequence

0 H0(XΣ,L′) H0(XΣ,L) H0(X3, L|X3)

H1(XΣ,L′) H1(XΣ,L) H1(X3, L|X3)

H2(XΣ,L′) H2(XΣ,L) H2(X3, L|X3)

α0 β0

δ0

α1

α2

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Hypersurface Case II

The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)

H0 (XΣ,L) H1 (XΣ,L) H2 (XΣ,L)

0 0 0

α0 α1 α2

Compute with Mathematica.

The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

0 0 0

H0(X3, L|X3

)H1(X3, L|X3

)

Vi := H i (XΣ,L) /Im(αi)

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The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)


0 0 0

α0 α1 α2


The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

0 0 0

H0(X3, L|X3

)H1(X3, L|X3

)


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The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)


0 0 0

α0 α1 α2


The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

0 0 0

H0(X3, L|X3

)

H1(X3, L|X3

)


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The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)


0 0 0

α0 α1 α2


The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

0 0 0

H0(X3, L|X3

)H1(X3, L|X3

)


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A Computational Example

Ingredients

Toric ambient space XΣ = CP2 × CP1 × CP1

s̃1 = C1x1 + C2x2 + C3x3 ∈ H0 (XΣ,OXΣ(1, 0, 0))

L = OXΣ(1, 0,−2)

Result

cohomCalg left an unconstraint constant A2 in the result.

My notebook computed this constant to be 0 forpseudo-random Ci ∈ (0, 1). So in this case

H0(XΣ, L|X3

)= 0, H1

(XΣ, L|X3

)= 2

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A Computational Example

Ingredients


s̃1 = C1x1 + C2x2 + C3x3 ∈ H0 (XΣ,OXΣ(1, 0, 0))

L = OXΣ(1, 0,−2)

Result

cohomCalg left an unconstraint constant A2 in the result.

My notebook computed this constant to be 0 forpseudo-random Ci ∈ (0, 1). So in this case

H0(XΣ, L|X3

)= 0, H1

(XΣ, L|X3

)= 2

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Questions?

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Towards Spectral SequencesOpen Question

Section 3

The Codimension Two Case

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The Task

XΣ

CY X3CY X3

Ba

Ingredients

Toric variety XΣ

L = OXΣ(D)

Polynomial s̃1 s.t.X3 = {s̃1 = 0}Polynomial s̃2 s.t.Ba = {s̃1 = s̃2 = 0}

Task

Compute H i(Ba, L|Ba

).

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The Task

XΣ

CY X3CY X3

Ba

Ingredients

Toric variety XΣ

L = OXΣ(D)


Task


).

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The Task

XΣ

CY X3

CY X3

Ba

Ingredients

Toric variety XΣ

L = OXΣ(D)

Polynomial s̃1 s.t.X3 = {s̃1 = 0}

Polynomial s̃2 s.t.Ba = {s̃1 = s̃2 = 0}

Task


).

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The Task

XΣ

CY X3CY X3

Ba

Ingredients

Toric variety XΣ

L = OXΣ(D)


Task


).

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The Task

XΣ

CY X3CY X3

Ba

Ingredients

Toric variety XΣ

L = OXΣ(D)


Task


).

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Codimension 2 Case

Theorem


0→ L′⊗

s̃2

−s̃1

−−−−−−−−→ V1

⊗(s̃1 ,̃s2)−−−−−→ L r→ L|Ba → 0

where

L′ = O (D − S1 − S2)

V1 = O (D − S1)⊕O (D − S2)

Watch out!

There is no associated long exact sequence in cohomology.

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Codimension 2 Case

Theorem


0→ L′⊗

s̃2

−s̃1

−−−−−−−−→ V1

⊗(s̃1 ,̃s2)−−−−−→ L r→ L|Ba → 0

where

L′ = O (D − S1 − S2)

V1 = O (D − S1)⊕O (D − S2)

Watch out!

There is no associated long exact sequence in cohomology.

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Spectral Sequence Construction Book by D. Cox, J. Little, H. Schenk ’toric varieties’,

’Aspects of (2,0) string compactifications’ by B. Green, J. Distler

Rough Picture

q

p

r

E 1,00

E 1,10

E 1,20

Ep,q0 are Abelian groups.

Derive E0 cohomologies.

Write those into sheet E1.

Derive E1 cohomologies.

. . .

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The Codimension 2 Strategy

The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)

H0 (XΣ,V1) H1 (XΣ,V1) H2 (XΣ,V1)


0 0 0

β0 β1 β2

α0 α1 α2

The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

Definitions

Vi := ker(βi)/im

(αi)Wi := H i (XΣ,L) /im

(βi)

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The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)

H0 (XΣ,V1) H1 (XΣ,V1) H2 (XΣ,V1)


0 0 0

β0 β1 β2

α0 α1 α2

The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

Definitions

Vi := ker(βi)/im


(βi)

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The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)

H0 (XΣ,V1) H1 (XΣ,V1) H2 (XΣ,V1)


0 0 0

β0 β1 β2

α0 α1 α2

The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

Definitions

Vi := ker(βi)/im


(βi)

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The E1-Sheet

0 0 0

H0 (XΣ,L′) H1 (XΣ,L′) H2 (XΣ,L′)

H0 (XΣ,V1) H1 (XΣ,V1) H2 (XΣ,V1)


0 0 0

β0 β1 β2

α0 α1 α2

The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

Definitions

Vi := ker(βi)/im


(βi)

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The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

α0(2) α1

(2)

The E3-Sheet

0 0 0

ker(α0)

ker(α1

(2)

)ker(α2

(2)

)

V0 V1 V2

W̃0 W̃1 W̃2

0 0 0

H0(Ba, L|Ba

)H1(Ba, L|Ba

)H2(Ba, L|Ba

)

Definition

W̃i := Wi/Im(αi

(2)

)

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The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

α0(2) α1

(2)

The E3-Sheet

0 0 0

ker(α0)

ker(α1

(2)

)ker(α2

(2)

)

V0 V1 V2

W̃0 W̃1 W̃2

0 0 0

H0(Ba, L|Ba

)H1(Ba, L|Ba

)H2(Ba, L|Ba

)

Definition

W̃i := Wi/Im(αi

(2)

)

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The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

α0(2) α1

(2)

The E3-Sheet

0 0 0

ker(α0)

ker(α1

(2)

)ker(α2

(2)

)

V0 V1 V2

W̃0 W̃1 W̃2

0 0 0

H0(Ba, L|Ba

)H1(Ba, L|Ba

)H2(Ba, L|Ba

)

Definition

W̃i := Wi/Im(αi

(2)

)15 / 20





The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

α0(2) α1

(2)

The E3-Sheet

0 0 0

ker(α0)

ker(α1

(2)

)ker(α2

(2)

)

V0 V1 V2

W̃0 W̃1 W̃2

0 0 0

H0(Ba, L|Ba

)

H1(Ba, L|Ba

)H2(Ba, L|Ba

)

Definition

W̃i := Wi/Im(αi

(2)

)15 / 20





The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

α0(2) α1

(2)

The E3-Sheet

0 0 0

ker(α0)

ker(α1

(2)

)ker(α2

(2)

)

V0 V1 V2

W̃0 W̃1 W̃2

0 0 0

H0(Ba, L|Ba

)H1(Ba, L|Ba

)

H2(Ba, L|Ba

)

Definition

W̃i := Wi/Im(αi

(2)

)15 / 20





The E2-Sheet

0 0 0

ker(α0)

ker(α1)

ker(α2)

V0 V1 V2

W0 W1 W2

0 0 0

α0(2) α1

(2)

The E3-Sheet

0 0 0

ker(α0)

ker(α1

(2)

)ker(α2

(2)

)

V0 V1 V2

W̃0 W̃1 W̃2

0 0 0

H0(Ba, L|Ba

)H1(Ba, L|Ba

)H2(Ba, L|Ba

)

Definition

W̃i := Wi/Im(αi

(2)

)15 / 20




Example With Knight’s Move I

Ingredients


S1 = (1, 0, 1) and S2 = (0, 0, 1)

L = OXΣ(1, 1, 0)

E2-Sheet

L′ 0 P1 0 0

V1 0 0 0 0

L P2 0 0 0

H0 H1 H2 H3

α0(2)

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Example With Knight’s Move I

Ingredients


S1 = (1, 0, 1) and S2 = (0, 0, 1)

L = OXΣ(1, 1, 0)

E2-Sheet

L′ 0 P1 0 0

V1 0 0 0 0

L P2 0 0 0

H0 H1 H2 H3

α0(2)

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Example With Knight’s Move II

Spaces And Polynomials

s̃1 = C4x1x5 + C2x2x5 + C3x1x6 + C1x2x6

s̃2 = C6x5 + C5x6

P1 ={A1 · x4

x5x6+ A2 · x3

x5x6, Ai ∈ C

}P2 = {A3 · x2x4 + A4 · x2x3 + A5x1x4 + A6x1x3 , Ai ∈ C}

It turns out that . . .

α0(2) : P1 → P2 is given by

α0(2) = x1x5x6 [C4C5 − C3C6] + x2x5x6 [C2C5 − C1C6]

Note

α0(2) respects the symmetries in P1 and P2.

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s̃1 = C4x1x5 + C2x2x5 + C3x1x6 + C1x2x6

s̃2 = C6x5 + C5x6

P1 ={A1 · x4

x5x6+ A2 · x3

x5x6, Ai ∈ C





Note


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s̃1 = C4x1x5 + C2x2x5 + C3x1x6 + C1x2x6

s̃2 = C6x5 + C5x6

P1 ={A1 · x4

x5x6+ A2 · x3

x5x6, Ai ∈ C





Note


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Open Questions

Fact hep-th/0808.3621, book by T. Huebsch ’Calabi-Yau Manifolds: A Bestiary for Physicists’

The cohomologies H i (CPn,L) are labeled by representationsof U (1)× U (n).

⇒ The cohomologies H i(CP1 × CP1 × CP1,L

)have

(anti)-symmetrisation properties.

Question To the definition

Is every smooth and compact normal toric variety XΣ a generalisedFlag variety?

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Open Questions

Fact hep-th/0808.3621, book by T. Huebsch ’Calabi-Yau Manifolds: A Bestiary for Physicists’

The cohomologies H i (CPn,L) are labeled by representationsof U (1)× U (n).

⇒ The cohomologies H i(CP1 × CP1 × CP1,L

)have

(anti)-symmetrisation properties.

Question To the definition

Is every smooth and compact normal toric variety XΣ a generalisedFlag variety?

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Section 4

Summary And Future Work

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Summary

Take-Away-Message

Computing the spectrum of massless zero modes requiresspectral sequence technology.

The existing Koszul extension of cohomCalg leavesunconstraint constants in these computations.

⇒ My notebook computes the E1-sheet and thereby fixes many,but not all, of these constants.

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Future Works

Open Tasks

Prove of disprove that every smooth and compact normaltoric variety XΣ is a generalised Flag variety.

Extend the functionality of the notebook beyondhypersurfaces.

Improve the performance of the notebook.

Apply the notebook to model building.

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Future Works

Open Tasks





20 / 20



Future Works

Open Tasks





20 / 20



Future Works

Open Tasks





20 / 20



Thank you for your attention! Questions?

20 / 20

Proof I

Claim

Let X a smooth and compact normal toric variety given by

XΣ∼= (Cr − Z ) / (C∗)a

We pick s̃1 ∈ H0 (XΣ,OXΣ(S1)) non-trivial and define

X3 = {p ∈ XΣ , s̃1 (p) = 0}

Then for any divisor class D ∈ Cl (XΣ) the following sequence issheaf exact

0→ OXΣ(D − S1)

⊗s̃1→ OXΣ(D)

r→ OXΣ(D)|X3

→ 0

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Proof II Back to orginal material.

Proof

Sheaf exactness is a local property. So let p ∈ XΣ a point.Then we have to show that the following sequence is exact

0→ OXΣ,p

[̃s1]p→ OXΣ,pr→ OXΣ,p/

([̃s1]p

)→ 0

Note that OXΣ,p is the local power series ring. This ring is anintegral domain. Therefore [̃s1]p is not a zero- divisor, so that

the map OXΣ,p

[̃s1]p→ OXΣ,p is injective.

The map OXΣ,pr→ OXΣ,p/

([̃s1]p

)is surjective.

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Generalised Flag Varieties To the conjecture

Definition

A simply connected, compact, complex, homogeneous G -space istermed a generalised Flag variety.

Example

It holdsCPn ∼= U (n + 1) / (U (1)× U (n))

Consequence

The cohomology groups H i (CPn,OCPn (k)) are labeled byrepresentations of U (1)× U (n) for all k ∈ Z.

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Definition


Example


Consequence


23 / 20


Definition


Example


Consequence


23 / 20

Documents

Cohomology Of Holomorphic Pullback Line Bundles On Smooth ... · Motivation From Physics The Hypersurface Case The Codimension Two Case Summary And Future Work Cohomology Of Holomorphic