Cohomological characterization of Universal bundles of the ... · Cohomological characterization of...
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Cohomological characterization of Universal bundles of the Grassmannian of lines Cohomological characterization of Universal bundles of the Grassmannian of lines Alicia Tocino S´ anchez Departamento de ´ Algebra Facultad de Ciencias Matem´ aticas, UCM Defensa de tesis para optar al grado de doctor en Matem´ aticas bajo la direcci´ on de Enrique Arrondo Esteban Madrid 2015
Transcript of Cohomological characterization of Universal bundles of the ... · Cohomological characterization of...
Cohomological characterization of Universal bundles of the Grassmannian of lines
Cohomological characterization of Universalbundles of the Grassmannian of lines
Alicia Tocino Sanchez
Departamento de AlgebraFacultad de Ciencias Matematicas, UCM
Defensa de tesis para optar al grado de doctor en Matematicasbajo la direccion de Enrique Arrondo Esteban
Madrid 2015
Cohomological characterization of Universal bundles of the Grassmannian of lines
Motivation
Horrock’s criterion A vector bundle F over Pn splits if andonly if F does not have intermediate cohomology
Ottaviani splitting criterion for G(k , n) and quadrics
Arrondo and Grana characterized ⊕O(li0)⊕
⊕Q(li1) (G(1, 4))
Costa and Miro-Roig characterized SλQ (G(k , n))
Arrondo and Malaspina gave an improvement of thesplitting criterion for the Grassmannian of lines
Cohomological characterization of Universal bundles of the Grassmannian of lines
1 Notation Q, S, universal exact sequence, E-N complex
2 Splitting criterion for Grassmannian of linesIdeas (Serre duality, E-N complex)ResultComparison
3 Next Step O, QIdeas (Induction on cohomology)Sketch of the proofResultComparison for G(1, 4)
4 General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)Ideas (Double induction)Sketch of the proofResult
5 Derived categories vs E-N complex + Serre Duality (Chapter 4)
Cohomological characterization of Universal bundles of the Grassmannian of lines
Notation Q, S, universal exact sequence, E-N complex
Notation for G(1, n):
G(1, n) = {1−dimensional subspaces of Pn = P(V )}
Q∨ = {(v ,Λ) ∈ V ∗ ×G(1, n) | v ∈ Λ}
The universal exact sequence is:
0 −→ Q∨ −→ V ∗ ⊗Oρ
−→ S −→ 0
Q∨ is the universal bundle of rank 2
S is the universal bundle of rank n − 1
By making ∧jρ we get some Eagon-Northcott complexes
Cohomological characterization of Universal bundles of the Grassmannian of lines
Splitting criterion for Grassmannian of lines
Ideas (Serre duality, E-N complex)
The following are equivalent:
O is a direct summand of F
there exist maps O −→ F and F −→ O whose composition isnon zero
We can relate the composition:
Hom(O,F ) × Hom(F ,O) −→ Hom(O,O)
(H0(F )× H0(F∨) −→ H0(O))
with the perfect pairing giving by Serre’s duality.
Cohomological characterization of Universal bundles of the Grassmannian of lines
Splitting criterion for Grassmannian of lines
Ideas (Serre duality, E-N complex)
We can get the following commutative diagram:
Hn−1(F ⊗ Sn−1Q(−n))× Hn−1(F∨ ⊗ Sn−1Q∨(−1))φ
−→ H2n−2(O(−n − 1))
↑ id × ψ2 ψ4 ↑≃
Hn−1(F ⊗ Sn−1Q(−n))× H0(F∨) −→ Hn−1(Sn−1Q(−n))
↑ ψ1 × id ψ3 ↑≃
H0(F )× H0(F∨)φ′
−→ H0(O)
Cohomological characterization of Universal bundles of the Grassmannian of lines
Splitting criterion for Grassmannian of lines
Ideas (Serre duality, E-N complex)
We use the following Eagon-Northcott complex to build theinjective map ψ1 : H
0(F ) −→ Hn−1(F ⊗ Sn−1Q(−n)):
0 → Sn−1Q(−n) → V ∗ ⊗ Sn−2Q(−n + 1) →∧2 V ∗ ⊗ Sn−3Q(−n + 2) → . . .
. . . →∧n−2 V ∗ ⊗Q(−2) →
∧n−1 V ∗ ⊗O(−1) → O → 0
And the following complex to build the surjective mapψ2 : H
0(F∨) −→ Hn−1(F∨ ⊗ Sn−1Q∨(−1)):
0 → Sn−1Q∨(−1) → V ∗ ⊗ Sn−2Q∨(−1) →∧2 V ∗ ⊗ Sn−3Q∨(−1) → . . .
. . . →∧n−2 V ∗ ⊗Q∨(−1) →
∧n−1 V ∗ ⊗O(−1) → O → 0
Cohomological characterization of Universal bundles of the Grassmannian of lines
Splitting criterion for Grassmannian of lines
Result
Step 0
(Splitting Criterion) A vector bundle F over G(1, n) splits if andonly if H j
∗(F ⊗ S iQ) = 0 where (i , j) ∈ A0 ∪ B0.
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B0
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Figure: AM’s splitting critetion
Cohomological characterization of Universal bundles of the Grassmannian of lines
Splitting criterion for Grassmannian of lines
Comparison
First we compare with the splitting criterion made by Ottaviani.
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2n − k − 2
2n − k − 1
...
2n − 7
2n − 6
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1 2 3 4 . . . k − 2 k − 1k k + 1 k + 2 . . . n − 5n − 4n − 3n − 2 i
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R0 =B0
L0 =A0
Figure: Ottaviani’s splitting criterion
Cohomological characterization of Universal bundles of the Grassmannian of lines
Splitting criterion for Grassmannian of lines
Comparison
Now we compare with the splitting criterion obtained by usingderived categories.
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1 2 3 4 5 6 . . . n − 5 n − 4 n − 3 n − 2 n − 1 i
M0 =A0
N0
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Figure: Splitting critetion with derived categories
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Ideas (Induction on cohomology)
Goal: Characterize direct sums of twists of O and Q.
As before the following are equivalent:
Q is a direct summand of F
there exist maps Q −→ F and F −→ Q whose composition isnon zero
We can relate the composition:
Hom(Q,F ) × Hom(F ,Q) −→ Hom(Q,Q)
(H0(F ⊗Q∨)× H0(F∨ ⊗Q) −→ H0(Q∨ ⊗Q))
with the perfect pairing giving by Serre’s duality.
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Ideas (Induction on cohomology)
We can get the following commutative diagram:
Hn−1(F ⊗ Sn−2Q(−n))× Hn−1(F∨ ⊗ Sn−2Q∨(−1))φ
−→ H2n−2(O(−n − 1))
↑ ψ1 × id ψ3 ↑≃
H0(F ⊗Q∨)× Hn−1(F∨ ⊗ Sn−2Q∨(−1)) −→ Hn−1(Q∨ ⊗ Sn−2Q∨(−1))
↑ id × ψ2 ψ4 ↑≃
H0(F ⊗Q∨)× H0(F∨ ⊗Q)φ′
−→ H0(Q∨ ⊗Q)
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Ideas (Induction on cohomology)
We can build the natural surjective maps ψ1 and ψ2 by using someparticular Eagon-Northcott complexes.
For ψ1 : H0(F ⊗Q∨) −→ Hn−1(F ⊗ Sn−2Q(−n)) we use:
0 −→ Sn−2Q(−n) −→ V ∗ ⊗ Sn−3Q(−n + 1) −→∧2 V ∗ ⊗ Sn−4Q(−n + 2) −→ . . .
. . . −→∧n−3 V ∗ ⊗Q(−3) −→
∧n−2 V ∗ ⊗O(−2) −→ V ⊗O(−1) −→ Q∨ −→ 0
And for ψ2 : H0(F∨ ⊗Q) −→ Hn−1(F∨ ⊗ Sn−2Q∨(−1)) we use:
0 −→ Sn−2Q∨(−1) −→ V ∗ ⊗ Sn−3Q∨(−1) −→∧2 V ∗ ⊗ Sn−4Q∨(−1) −→ . . .
. . . −→∧n−3 V ∗ ⊗Q∨(−1) −→
∧n−2 V ∗ ⊗O(−1) −→ V ⊗O −→ Q −→ 0
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Ideas (Induction on cohomology)
If we only suppose the vanishing of cohomology that make ψ1 andψ2 surjective maps we are just characterizing direct sums of twistsof Q. The hypotheses are H j
∗(F ⊗ S iQ) = 0 with (i , j) in thefollowing figure.
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n − 3
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2n − 6
2n − 5
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2n − 3
2n − 2
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1 2 3 4 . . . n − 5 n − 4 n − 3 n − 2 n − 1 i
C1
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Figure: Characterize direct sums of twists of Q
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Sketch of the proof
Ideas to characterize direct sums of twists of O and Q:
Remove one hypothesis of Step 0 (Hn−1∗ (F ⊗ Sn−2Q) 6= 0)
Our conditions: the remaining hypotheses of Step 0 and thehypotheses that characterize direct sums of twists of Q
Induction on∑
l hn−1(F ⊗ Sn−2Q(l)) = m.
m = 0 ⇒ Step 0we suppose the result true for m − 1we prove the result for m 6= 0
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Sketch of the proof
Sketch of the proof:
There exists an l such that Hn−1(F ⊗ Sn−2Q(l)) 6= 0 (wechoose l = −n)
We obtain the commutative diagram ⇒ F has as a directsummand Q:
F = Q⊕ F ′
F ′ satisfies the same hypotheses of F and
m′ :=∑
l
hn−1(F ′ ⊗ Sn−2Q(l)) = m − 1
Applying induction hypothesis to F ′ ⇒ F can be expressed asdirect sums of twists of O and Q
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Result
Step 1
A vector bundle F over G(1, n) can be expressed as direct sums oftwists of O and Q if and only if H j
∗(F ⊗ S iQ) = 0 where (i , j) arethe points in the following figure.
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2n − 6
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2n − 3
2n − 2
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1 2 3 4 . . . n − 5 n − 4 n − 3 n − 2 n − 1 i
C1
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Figure: Characterize direct sums of twists of O and Q
Cohomological characterization of Universal bundles of the Grassmannian of lines
Next Step O, Q
Comparison for G(1, 4)
We can compare this result with the one given by E. Arrondo andB. Grana for the case G(1, 4). Observe that our characterizationhas one less condition.
A vector bundle F over G(1, 4) can be expressed as direct sums oftwists of O and Q if and only if H j
∗(F ⊗ S iQ) = 0 where thepoints (i , j) are in the following figure.
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Figure: AG characterization vs Step 1
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Ideas (Double induction)
Goal: Characterize direct sums of twists of O, Q, S2Q, . . ., SkQwith k ≤ n − 2.
The following are equivalent:
SkQ is a direct summand of F
there exist maps SkQ −→ F and F −→ SkQ whosecomposition is non zero
We can relate the composition:Hom(SkQ,F )× Hom(F ,SkQ) → Hom(SkQ,SkQ)
(H0(F ⊗ SkQ∨)× H0(F∨ ⊗ SkQ) → H0(SkQ∨ ⊗ SkQ))with the perfect pairing giving by Serre’s duality.
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Ideas (Double induction)
We can get the following commutative diagram:
Hn−1(F ⊗ Sn−k−1Q(−n))× Hn−1(F∨ ⊗ Sn−k−1Q∨(−1))φ
−→ H2n−2(O(−n − 1))
↑ ψ1 × id ψ3 ↑≃
H0(F ⊗ SkQ∨)× Hn−1(F∨ ⊗ Sn−k−1Q∨(−1)) −→ Hn−1(SkQ∨ ⊗ Sn−k−1Q∨(−1))
↑ id × ψ2 ψ4 ↑≃
H0(F ⊗ SkQ∨)× H0(F∨ ⊗ SkQ)φ′
−→ H0(SkQ∨ ⊗ SkQ)
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Ideas (Double induction)
We can build the natural surjective maps ψ1 and ψ2 by using someparticular Eagon-Northcott complexes.
For ψ1 : H0(F ⊗ SkQ∨) −→ Hn−1(F ⊗ Sn−k−1Q(−n)) we use:
0 → Sn−k−1Q(−n) → V ∗ ⊗ Sn−k−2Q(−n + 1) →∧2 V ∗ ⊗ Sn−k−3Q(−n + 2) → . . .
. . . →∧n−k−2 V ∗ ⊗Q(−k − 2) →
∧n−k−1 V ∗ ⊗O(−k − 1) →
→∧k V ⊗O(−k) →
∧k−1 V ⊗Q(−k) → . . .
. . . →∧2 V ⊗ Sk−2Q(−k) → V ⊗ Sk−1Q(−k) → SkQ(−k) → 0
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Ideas (Double induction)
And for ψ2 : H0(F∨ ⊗ SkQ) −→ Hn−1(F∨ ⊗ Sn−k−1Q∨(−1)) we
use:
0 → Sn−k−1Q∨(−1) → V ∗ ⊗ Sn−k−2Q∨(−1) →∧2 V ∗ ⊗ Sn−k−3Q∨(−1) → . . .
. . . →∧n−k−2 V ∗ ⊗Q∨(−1) →
∧n−k−1 V ∗ ⊗O(−1) →
→∧k V ⊗O →
∧k−1 V ⊗Q → . . .
. . . →∧2 V ⊗ Sk−2Q → V ⊗ Sk−1Q → SkQ → 0
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Ideas (Double induction)
If we only suppose the vanishing of cohomology that make ψ1 andψ2 surjective maps we are just characterizing direct sums of twistsof SkQ. The hypotheses are H j
∗(F ⊗ S iQ) = 0 with (i , j) in thefollowing figure.
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...
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2n − k
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2n − 4
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Ck
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1 2 3 . . . k − 3 k − 2 k − 1 . . . n − k − 5n − k − 4
n − k − 3n − k − 2
. . . . . . n − 3 n − 2 n − 1 i
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Figure: Characterize direct sums of twists of SkQ
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Sketch of the proof
Ideas to characterize direct sums of O, Q, S2Q, . . ., SkQ:
Remove one hypothesis of Step k-1(Hn−1
∗ (F ⊗ Sn−k−1Q) 6= 0)
Our conditions: the remaining hypotheses of Step k − 1 andthe hypotheses that characterize direct sums of twists of SkQ
Induction on∑
l hn−1(F ⊗ Sn−k−1Q(l)) = m.
m = 0 ⇒ Step k − 1we suppose the result true for m − 1we prove the result for m 6= 0
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Sketch of the proof
Sketch of the proof:
There exists an l such that Hn−1(F ⊗ Sn−k−1Q(l)) 6= 0 (wechoose l = −n)
We obtain the commutative diagram ⇒ F has as a directsummand SkQ:
F = SkQ⊕ F ′
F ′ satisfies the same hypotheses of F and
m′ :=∑
l
hn−1(F ′ ⊗ Sn−k−1Q(l)) = m − 1
Applying induction hypothesis to F ′ ⇒ F can be expressed asdirect sums of twists of O, Q, S2Q, . . ., Sk−1Q and SkQ
Cohomological characterization of Universal bundles of the Grassmannian of lines
General Step O, Q, S2Q, . . ., SkQ with k ≤ n − 2 (Chapter 3)
Result
Step k
A vector bundle F over G(1, n) can be expressed as direct sums oftwists of O, Q, S2Q, . . ., Sk−1Q and SkQ with k ≤ n − 2 if andonly if H j
∗(F ⊗ S iQ) = 0 where (i , j) are the points in the followingfigure.
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2n − k − 1
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b(n − k − 2, n − 1)
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n − k − 2 . . . . . . n − 4 n − 3 n − 2 n − 1
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Figure: Characterize direct sums of twists of O, Q, S2Q, . . ., SkQ
Cohomological characterization of Universal bundles of the Grassmannian of lines
Derived categories vs E-N complex + Serre Duality (Chapter 4)
To use derived categories we consider the following resolution ofthe diagonal ∆ ⊆ X × X where X = G(k , n):
0 →∧(k+1)(n−k)(Q∨ ⊠ S∨) → . . .
∧2(Q∨ ⊠ S∨) → Q∨ ⊠ S∨ → OX×X → O∆ → 0
These elements decompose in the following way:
r∧(Q∨⊠ S∨) =
⊕
|λ|=r
SλQ∨⊠ Sλ′S∨
where the sum goes over all Young tableau with r cells, λ′ is theconjugate Young tableau and Sλ is the Schur functor associated tothe tableau λ
Cohomological characterization of Universal bundles of the Grassmannian of lines
Derived categories vs E-N complex + Serre Duality (Chapter 4)
Observe that H2n−2(O(−n − 1)) = Ext2n−2(O,O(−n − 1)) andthe element that generates is precisely:
0 −→ O(−n− 1) −→∧n−1 V ⊗O(−n − 2) −→
∧n−2 V ⊗Q(−n) −→ . . .
. . . −→∧2 V ⊗ Sn−3Q(−n) −→ V ⊗ Sn−2Q(−n) −→ V ∗ ⊗ Sn−2Q(−n + 1) −→ . . .
. . . −→∧n−2 V ∗ ⊗Q(−2) −→
∧n−1 V ∗ ⊗O(−1) −→ O −→ 0
Conclusion:
The resolution of the diagonal has more pieces than the previouscomplex.
Cohomological characterization of Universal bundles of the Grassmannian of lines
Derived categories vs E-N complex + Serre Duality (Chapter 4)
Thank you!
