Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf ·...
Transcript of Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf ·...
![Page 1: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/1.jpg)
Index Formulae for Line Bundle Cohomologyon Complex Surfaces
Callum BrodieUniversity of Oxford
Based on 1906.08363, 1906.08730, and 1906.08769
with Andrei Constantin, Rehan Deen, and Andre Lukas
27th of June 2019
![Page 2: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/2.jpg)
Where will we end up?
Understand how for many surfaces, e.g. all compact toric,
h0(S,L) = χ(L̃) ∀L
And on dPn and Fn go further: closed-form expression for h0.
Understand cohomology structure illustrated in these pictures:
![Page 3: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/3.jpg)
Where will we end up?
Understand how for many surfaces, e.g. all compact toric,
h0(S,L) = χ(L̃) ∀L
And on dPn and Fn go further: closed-form expression for h0.
Understand cohomology structure illustrated in these pictures:
![Page 4: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/4.jpg)
Where will we end up?
Understand how for many surfaces, e.g. all compact toric,
h0(S,L) = χ(L̃) ∀L
And on dPn and Fn go further: closed-form expression for h0.
Understand cohomology structure illustrated in these pictures:
![Page 5: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/5.jpg)
Where will we end up?
Understand how for many surfaces, e.g. all compact toric,
h0(S,L) = χ(L̃) ∀L
And on dPn and Fn go further: closed-form expression for h0.
Understand cohomology structure illustrated in these pictures:
![Page 6: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/6.jpg)
Motivation
Andrei’s preceding talk motivated understandingformulae for line bundle cohomology, but . . .
Why (complex) surfaces?
• Surfaces are building blocks for CYs(toric, del Pezzos, Hirzebruchs . . . )
⇒ cohomology directly useful for e.g. model-building
• Simpler arena to understand cohomology
⇒ learn about CY3 cohomology structure too?
![Page 7: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/7.jpg)
Motivation
Andrei’s preceding talk motivated understandingformulae for line bundle cohomology, but . . .
Why (complex) surfaces?
• Surfaces are building blocks for CYs(toric, del Pezzos, Hirzebruchs . . . )
⇒ cohomology directly useful for e.g. model-building
• Simpler arena to understand cohomology
⇒ learn about CY3 cohomology structure too?
![Page 8: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/8.jpg)
Motivation
Andrei’s preceding talk motivated understandingformulae for line bundle cohomology, but . . .
Why (complex) surfaces?
• Surfaces are building blocks for CYs(toric, del Pezzos, Hirzebruchs . . . )
⇒ cohomology directly useful for e.g. model-building
• Simpler arena to understand cohomology
⇒ learn about CY3 cohomology structure too?
![Page 9: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/9.jpg)
Line bundles and divisors
Line bundles in one-to-one correspondence with divisors,
L ≡ O(D) ,
up to linear equivalence.
Particularly simple connection for zeroth cohomology,
h0(L) = dim (|D|def) + 1 .
where |D|def is the space of deformations of D, or morecorrectly the complete linear system.
Key idea for us: some divisor parts can be rigid - always incomplete linear system, don’t contribute to deformations.
![Page 10: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/10.jpg)
Line bundles and divisors
Line bundles in one-to-one correspondence with divisors,
L ≡ O(D) ,
up to linear equivalence.
Particularly simple connection for zeroth cohomology,
h0(L) = dim (|D|def) + 1 .
where |D|def is the space of deformations of D, or morecorrectly the complete linear system.
Key idea for us: some divisor parts can be rigid - always incomplete linear system, don’t contribute to deformations.
![Page 11: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/11.jpg)
Line bundles and divisors
Line bundles in one-to-one correspondence with divisors,
L ≡ O(D) ,
up to linear equivalence.
Particularly simple connection for zeroth cohomology,
h0(L) = dim (|D|def) + 1 .
where |D|def is the space of deformations of D, or morecorrectly the complete linear system.
Key idea for us: some divisor parts can be rigid - always incomplete linear system, don’t contribute to deformations.
![Page 12: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/12.jpg)
Line bundles and divisors
Line bundles in one-to-one correspondence with divisors,
L ≡ O(D) ,
up to linear equivalence.
Particularly simple connection for zeroth cohomology,
h0(L) = dim (|D|def) + 1 .
where |D|def is the space of deformations of D, or morecorrectly the complete linear system.
Key idea for us: some divisor parts can be rigid - always incomplete linear system, don’t contribute to deformations.
![Page 13: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/13.jpg)
Rough summary of idea
Dropping rigid pieces doesn’t affect deformations, so zerothcohomology of associated bundle is unaffected,
h0(S,OS(D)) = h0(S,OS(D −Drigid)) ,
when Drigid is a fixed piece in deformations |D|def .
Why is this useful? If we throw away enough rigid pieces,often resulting bundle has simpler cohomology, specifically
h0(S,OS(D −Drigid))often= χ(S,OS(D −Drigid)) .
But this is only useful in practice if we can detect rigid pieces.Happily, rigid pieces can be detected by intersections,
D ·Drigid < 0 ⇒ Drigid ⊂ |D|def .
![Page 14: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/14.jpg)
Rough summary of idea
Dropping rigid pieces doesn’t affect deformations, so zerothcohomology of associated bundle is unaffected,
h0(S,OS(D)) = h0(S,OS(D −Drigid)) ,
when Drigid is a fixed piece in deformations |D|def .
Why is this useful? If we throw away enough rigid pieces,often resulting bundle has simpler cohomology, specifically
h0(S,OS(D −Drigid))often= χ(S,OS(D −Drigid)) .
But this is only useful in practice if we can detect rigid pieces.Happily, rigid pieces can be detected by intersections,
D ·Drigid < 0 ⇒ Drigid ⊂ |D|def .
![Page 15: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/15.jpg)
Rough summary of idea
Dropping rigid pieces doesn’t affect deformations, so zerothcohomology of associated bundle is unaffected,
h0(S,OS(D)) = h0(S,OS(D −Drigid)) ,
when Drigid is a fixed piece in deformations |D|def .
Why is this useful? If we throw away enough rigid pieces,often resulting bundle has simpler cohomology, specifically
h0(S,OS(D −Drigid))often= χ(S,OS(D −Drigid)) .
But this is only useful in practice if we can detect rigid pieces.Happily, rigid pieces can be detected by intersections,
D ·Drigid < 0 ⇒ Drigid ⊂ |D|def .
![Page 16: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/16.jpg)
Rough summary of idea
Dropping rigid pieces doesn’t affect deformations, so zerothcohomology of associated bundle is unaffected,
h0(S,OS(D)) = h0(S,OS(D −Drigid)) ,
when Drigid is a fixed piece in deformations |D|def .
Why is this useful? If we throw away enough rigid pieces,often resulting bundle has simpler cohomology, specifically
h0(S,OS(D −Drigid))often= χ(S,OS(D −Drigid)) .
But this is only useful in practice if we can detect rigid pieces.Happily, rigid pieces can be detected by intersections,
D ·Drigid < 0 ⇒ Drigid ⊂ |D|def .
![Page 17: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/17.jpg)
Divisor intersections
Intersection theory has negative self-intersections, (D−)2 = 0.For example exceptional blow-up divisors.
If a divisor D contains D−as a part, then there arenegative contributions to intersection (D−) ·D.
But intersection theory is defined up to equivalence:so doesn’t care about individual deformations.
So negative intersection (D−) ·D < 0 meansevery deformation contains the piece D−.
![Page 18: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/18.jpg)
Divisor intersections
Intersection theory has negative self-intersections, (D−)2 = 0.For example exceptional blow-up divisors.
If a divisor D contains D−as a part, then there arenegative contributions to intersection (D−) ·D.
But intersection theory is defined up to equivalence:so doesn’t care about individual deformations.
So negative intersection (D−) ·D < 0 meansevery deformation contains the piece D−.
![Page 19: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/19.jpg)
Divisor intersections
Intersection theory has negative self-intersections, (D−)2 = 0.For example exceptional blow-up divisors.
If a divisor D contains D−as a part, then there arenegative contributions to intersection (D−) ·D.
But intersection theory is defined up to equivalence:so doesn’t care about individual deformations.
So negative intersection (D−) ·D < 0 meansevery deformation contains the piece D−.
![Page 20: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/20.jpg)
Divisor intersections
Intersection theory has negative self-intersections, (D−)2 = 0.For example exceptional blow-up divisors.
If a divisor D contains D−as a part, then there arenegative contributions to intersection (D−) ·D.
But intersection theory is defined up to equivalence:so doesn’t care about individual deformations.
So negative intersection (D−) ·D < 0 meansevery deformation contains the piece D−.
![Page 21: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/21.jpg)
Divisor intersections
Intersection theory has negative self-intersections, (D−)2 = 0.For example exceptional blow-up divisors.
If a divisor D contains D−as a part, then there arenegative contributions to intersection (D−) ·D.
But intersection theory is defined up to equivalence:so doesn’t care about individual deformations.
So negative intersection (D−) ·D < 0 meansevery deformation contains the piece D−.
![Page 22: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/22.jpg)
General theorems
Question:How many rigid pieces can bedetected with intersections?
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General theorems
TheoremLet D be an effective divisor on a smooth compact complex projectivesurface S, with associated line bundle OS(D). Let I be the set ofirreducible negative self-intersection divisors. Then the following map,
D → D̃ = D −∑C∈I
θ(−D · C) ceil
(D · CC2
)C ,
where θ is the step function, preserves the zeroth cohomology,
h0(S,OS(D̃)
)= h0
(S,OS(D)
).
CorollaryWrite D̃ for the divisor that is the result of iterating the map D → D̃,until stabilisation after a finite number of steps. Then D̃ is a nefdivisor such that h0
(S,O(D)
)= h0
(S,O(D̃)
).
![Page 24: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/24.jpg)
General theorems
TheoremLet D be an effective divisor on a smooth compact complex projectivesurface S, with associated line bundle OS(D). Let I be the set ofirreducible negative self-intersection divisors. Then the following map,
D → D̃ = D −∑C∈I
θ(−D · C) ceil
(D · CC2
)C ,
where θ is the step function, preserves the zeroth cohomology,
h0(S,OS(D̃)
)= h0
(S,OS(D)
).
CorollaryWrite D̃ for the divisor that is the result of iterating the map D → D̃,until stabilisation after a finite number of steps. Then D̃ is a nefdivisor such that h0
(S,O(D)
)= h0
(S,O(D̃)
).
![Page 25: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/25.jpg)
Combination with vanishing theorems
Divisor D mapped to new divisor D̃: D → D̃ → . . .→ D̃.
If higher cohomologies vanish for D̃, then h0 reduces toindex computation (simpler, topological),
if h1(S,OS(D̃)) = h2(S,OS(D̃)) = 0 ,
then h0(S,OS(D)) = h0(S,OS(D̃)) = χ(S,OS(D̃)) .
Can we make general statements on vanishing for D̃?Yes when there are vanishing theorems.
CorollaryIf a vanishing theorem ensures that hq(S,L) = 0 for q > 0 for all nefbundles L, then all zeroth cohomology is given by an index,
h0(S,OS(D)) = χ(S,OS(D̃)) .
![Page 26: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/26.jpg)
Combination with vanishing theorems
Divisor D mapped to new divisor D̃: D → D̃ → . . .→ D̃.
If higher cohomologies vanish for D̃, then h0 reduces toindex computation (simpler, topological),
if h1(S,OS(D̃)) = h2(S,OS(D̃)) = 0 ,
then h0(S,OS(D)) = h0(S,OS(D̃)) = χ(S,OS(D̃)) .
Can we make general statements on vanishing for D̃?Yes when there are vanishing theorems.
CorollaryIf a vanishing theorem ensures that hq(S,L) = 0 for q > 0 for all nefbundles L, then all zeroth cohomology is given by an index,
h0(S,OS(D)) = χ(S,OS(D̃)) .
![Page 27: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/27.jpg)
Combination with vanishing theorems
Divisor D mapped to new divisor D̃: D → D̃ → . . .→ D̃.
If higher cohomologies vanish for D̃, then h0 reduces toindex computation (simpler, topological),
if h1(S,OS(D̃)) = h2(S,OS(D̃)) = 0 ,
then h0(S,OS(D)) = h0(S,OS(D̃)) = χ(S,OS(D̃)) .
Can we make general statements on vanishing for D̃?Yes when there are vanishing theorems.
CorollaryIf a vanishing theorem ensures that hq(S,L) = 0 for q > 0 for all nefbundles L, then all zeroth cohomology is given by an index,
h0(S,OS(D)) = χ(S,OS(D̃)) .
![Page 28: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/28.jpg)
Combination with vanishing theorems
Divisor D mapped to new divisor D̃: D → D̃ → . . .→ D̃.
If higher cohomologies vanish for D̃, then h0 reduces toindex computation (simpler, topological),
if h1(S,OS(D̃)) = h2(S,OS(D̃)) = 0 ,
then h0(S,OS(D)) = h0(S,OS(D̃)) = χ(S,OS(D̃)) .
Can we make general statements on vanishing for D̃?Yes when there are vanishing theorems.
CorollaryIf a vanishing theorem ensures that hq(S,L) = 0 for q > 0 for all nefbundles L, then all zeroth cohomology is given by an index,
h0(S,OS(D)) = χ(S,OS(D̃)) .
![Page 29: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/29.jpg)
Example: Compact toric surfaces
For the important class of compact toric surfaces there is apowerful vanishing theorem (Demazure) for nef bundles.
CorollaryLet S be a compact toric surface, and D an effective divisor. Then
h0(S,O(D)
)= χ(D̃) ,
where the divisor D̃ is obtained from D by iterating the shifts D → D̃.
![Page 30: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/30.jpg)
Example: Compact toric surfaces
For the important class of compact toric surfaces there is apowerful vanishing theorem (Demazure) for nef bundles.
CorollaryLet S be a compact toric surface, and D an effective divisor. Then
h0(S,O(D)
)= χ(D̃) ,
where the divisor D̃ is obtained from D by iterating the shifts D → D̃.
![Page 31: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/31.jpg)
Single-shift cases
If there is a vanishing theorem for the nef cone,and D → D̃ → . . .→ D̃ stabilises after one step,then we have an immediate closed-form expression for h0.
CorollaryIf D̃ is always nef and hq(SS ,OS(D̃)) = 0 for q > 0 for all D̃, then foreffective D we have the closed-form expression,
h0(S,OS(D)) = χ
(D −
∑C∈I
θ(−D · C) ceil
(D · CC2
)C
).
Surprisingly, there are many interesting examples of suchsurfaces, including Hirzebruch and del Pezzo surfaces.
![Page 32: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/32.jpg)
Single-shift cases
If there is a vanishing theorem for the nef cone,and D → D̃ → . . .→ D̃ stabilises after one step,then we have an immediate closed-form expression for h0.
CorollaryIf D̃ is always nef and hq(SS ,OS(D̃)) = 0 for q > 0 for all D̃, then foreffective D we have the closed-form expression,
h0(S,OS(D)) = χ
(D −
∑C∈I
θ(−D · C) ceil
(D · CC2
)C
).
Surprisingly, there are many interesting examples of suchsurfaces, including Hirzebruch and del Pezzo surfaces.
![Page 33: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/33.jpg)
Single-shift cases
If there is a vanishing theorem for the nef cone,and D → D̃ → . . .→ D̃ stabilises after one step,then we have an immediate closed-form expression for h0.
CorollaryIf D̃ is always nef and hq(SS ,OS(D̃)) = 0 for q > 0 for all D̃, then foreffective D we have the closed-form expression,
h0(S,OS(D)) = χ
(D −
∑C∈I
θ(−D · C) ceil
(D · CC2
)C
).
Surprisingly, there are many interesting examples of suchsurfaces, including Hirzebruch and del Pezzo surfaces.
![Page 34: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/34.jpg)
Example: Hirzebruch surfaces Fn
Vanishing theorem for nef bundles (Demazure)Shift D → D̃ terminates in one step
h0 (Fn,OFn(D)) = χ
(D − θ(−D · C) ceil
(D · CC2
)C
).
where C is the unique negative self-intersection curve.
Figure showssituation for F2
(2d Picard lattice)
Region 1 is the nef cone
![Page 35: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/35.jpg)
Example: Hirzebruch surfaces Fn
Vanishing theorem for nef bundles (Demazure)Shift D → D̃ terminates in one step
h0 (Fn,OFn(D)) = χ
(D − θ(−D · C) ceil
(D · CC2
)C
).
where C is the unique negative self-intersection curve.
Figure showssituation for F2
(2d Picard lattice)
Region 1 is the nef cone
![Page 36: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/36.jpg)
Example: Hirzebruch surfaces Fn
Vanishing theorem for nef bundles (Demazure)Shift D → D̃ terminates in one step
h0 (Fn,OFn(D)) = χ
(D − θ(−D · C) ceil
(D · CC2
)C
).
where C is the unique negative self-intersection curve.
Figure showssituation for F2
(2d Picard lattice)
Region 1 is the nef cone
![Page 37: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/37.jpg)
Example: del Pezzo surfaces dPn
Vanishing theorem for nef bundles (Kawamata-Viehweg)Shift D → D̃ terminates in one step
h0 (dPn,OdPn(D)) = χ
(D +
∑Ci
θ(−D · Ci) (D · Ci)Ci
),
where Ci are the exceptional curves.
Figure showssituation for dP2
(3d Picard lattice)
Region 1 is the nef cone
![Page 38: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/38.jpg)
Example: del Pezzo surfaces dPn
Vanishing theorem for nef bundles (Kawamata-Viehweg)Shift D → D̃ terminates in one step
h0 (dPn,OdPn(D)) = χ
(D +
∑Ci
θ(−D · Ci) (D · Ci)Ci
),
where Ci are the exceptional curves.
Figure showssituation for dP2
(3d Picard lattice)
Region 1 is the nef cone
![Page 39: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/39.jpg)
Example: del Pezzo surfaces dPn
Vanishing theorem for nef bundles (Kawamata-Viehweg)Shift D → D̃ terminates in one step
h0 (dPn,OdPn(D)) = χ
(D +
∑Ci
θ(−D · Ci) (D · Ci)Ci
),
where Ci are the exceptional curves.
Figure showssituation for dP2
(3d Picard lattice)
Region 1 is the nef cone
![Page 40: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/40.jpg)
Results• Understanding of structure for h0 for surfaces.• For compact toric surfaces: algorithm to get h0 as index.• For dPn and Fn: closed-form index formulae for h0.
Applications• Use surfaces as building blocks for CY3 and lift h0 to CY3.
⇒ e.g. proof of formulae for all h0 for many elliptic CY3.• Can reverse-engineer rigid divisors from cohomology.
(See Andre’s plenary talk.)
Extensions• Extend to other surfaces? K3, general type, . . .• Extend proofs to higher dimensions? 3-folds, 4-folds, . . .• Extend results to higher cohomologies? h1, h2, . . .
![Page 41: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/41.jpg)
Results• Understanding of structure for h0 for surfaces.• For compact toric surfaces: algorithm to get h0 as index.• For dPn and Fn: closed-form index formulae for h0.
Applications• Use surfaces as building blocks for CY3 and lift h0 to CY3.
⇒ e.g. proof of formulae for all h0 for many elliptic CY3.• Can reverse-engineer rigid divisors from cohomology.
(See Andre’s plenary talk.)
Extensions• Extend to other surfaces? K3, general type, . . .• Extend proofs to higher dimensions? 3-folds, 4-folds, . . .• Extend results to higher cohomologies? h1, h2, . . .
![Page 42: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/42.jpg)
Results• Understanding of structure for h0 for surfaces.• For compact toric surfaces: algorithm to get h0 as index.• For dPn and Fn: closed-form index formulae for h0.
Applications• Use surfaces as building blocks for CY3 and lift h0 to CY3.
⇒ e.g. proof of formulae for all h0 for many elliptic CY3.• Can reverse-engineer rigid divisors from cohomology.
(See Andre’s plenary talk.)
Extensions• Extend to other surfaces? K3, general type, . . .• Extend proofs to higher dimensions? 3-folds, 4-folds, . . .• Extend results to higher cohomologies? h1, h2, . . .
![Page 43: Index FormulaeforLine Bundle Cohomology onComplex Surfaces › ... › 3078385 › 4_Brodie.pdf · irreducible negative self-intersection divisors. Then the following map, D!D~ =](https://reader033.fdocument.org/reader033/viewer/2022060415/5f1338d738d7480bc567d644/html5/thumbnails/43.jpg)
Thank you for your attention