Intersections of two Grassmannians in ℙ9

Lev A. Borisov; Andrei Căldăraru; Alexander Perry

doi:10.1515/crelle-2018-0014

Published by De Gruyter May 29, 2018

Intersections of two Grassmannians in ℙ⁹

Lev A. Borisov , Andrei Căldăraru and Alexander Perry

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2018-0014

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Abstract

We study the intersection of two copies of Gr⁢(2,5) embedded in ℙ9, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi–Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi–Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi–Yau threefolds of the above type, which may be of independent interest.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1201466

Award Identifier / Grant number: DMS-1601907

Award Identifier / Grant number: DMS-1200721

Award Identifier / Grant number: DMS-1606460

Funding statement: Lev A. Borisov was partially supported by NSF Grants DMS-1201466 and DMS-1601907. Andrei Căldăraru was partially supported by NSF grant DMS-1200721. Alexander Perry was partially supported by an NSF postdoctoral fellowship, DMS-1606460.

A Borel–Weil–Bott computations

The purpose of this appendix is to collect some coherent cohomology computations on Grassmannians and GPK3 threefolds, which are invoked in the main text. The key tool is Borel–Weil–Bott, which we review in Section A.1.

A.1 Borel–Weil–Bott

For this subsection, we let V denote an n-dimensional vector space over k (in the rest of the paper n=5). The Borel–Weil–Bott Theorem for GL⁢(V) allows us to compute the coherent cohomology of GL⁢(V)-equivariant bundles on a Grassmannian Gr⁢(r,V) (in the rest of the paper we only need the case r=2). To state the result, we need some notation. Our exposition follows [19, Section 2.6].

The weight lattice of GL⁢(V) is isomorphic to ℤn via the map taking the d-th fundamental weight, i.e. the highest weight of ⋀dV, to the sum of the first d basis vectors of ℤn. Under this isomorphism, the dominant integral weights of GL⁢(V) correspond to nonincreasing sequences of integers λ=(λ1,…,λn). For such a λ, we denote by Σλ⁢V the corresponding irreducible representation of GL⁢(V) of highest weight λ. The only facts we shall need about these representations are the following:

If λ=(1,…,1,0,…,0) with the first d entries equal to 1, then Σλ⁢V=⋀dV.
If λ=(λ1,…,λn) and μ=(λ1+m,…,λn+m) for some m∈ℤ, then there is an isomorphism of GL⁢(V)-representations Σμ⁢V≅Σλ⁢V⊗det⁡(V)⊗m.
Given λ=(λ1,…,λn), set λ∨=(-λn,-λn-1,…,-λ1). Then there is an isomorphism of GL⁢(V)-representations Σλ∨⁢V≅(Σλ⁢V)∨.

The construction V↦Σλ⁢V for a dominant integral weight λ globalizes to vector bundles over a scheme, and the above identities continue to hold. We are interested in the case where the base scheme is the Grassmannian Gr⁢(r,V). Denote by 𝒰 the tautological rank r bundle on Gr⁢(r,V), and by 𝒬 the rank n-r quotient of V⊗𝒪Gr⁢(r,V) by 𝒰, so that there is an exact sequence

0→𝒰→V⊗𝒪→𝒬→0.

Then every GL⁢(V)-equivariant bundle on Gr⁢(r,V) is of the form Σα⁢𝒰∨⊗Σβ⁢𝒬∨ for some nonincreasing sequences of integers α∈ℤr and β∈ℤn-r.

The symmetric group Sn acts on the weight lattice ℤn by permuting the factors. Denote by ℓ:Sn→ℤ the standard length function. We say λ∈ℤn is regular if all of its components are distinct; in this case, there is a unique σ∈Sn such that σ⁢(λ) is a strictly decreasing sequence. Finally, let

ρ=(n,n-1,…,2,1)∈ℤn

be the sum of the fundamental weights.

The following result can be deduced from the usual statement of Borel–Weil–Bott by pushing forward equivariant line bundles on the flag variety to the Grassmannian. For a vector space L and an integer p, we write L⁢[p] for the single-term complex of vector spaces with L in degree -p.

Proposition A.1.

Let the notation be as above. Let α∈Zr and β∈Zn-r be nonincreasing sequences of integers, and let λ=(α,β)∈Zn be their concatenation. If λ+ρ is not regular, then

R⁢Γ⁢(Gr⁢(r,V),Σα⁢𝒰∨⊗Σβ⁢𝒬∨)≅0.

If λ+ρ is regular and σ∈Sn is the unique element such that σ⁢(λ+ρ) is a strictly decreasing sequence, then

R⁢Γ⁢(Gr⁢(r,V),Σα⁢𝒰∨⊗Σβ⁢𝒬∨)≅Σσ⁢(λ+ρ)-ρ⁢V∨⁢[-ℓ⁢(σ)].

For r=2 we express the normal bundle of the Plücker embedding Gr⁢(2,V)⊂ℙ⁢(⋀2V) in a form that is well suited to applying Proposition A.1.

Lemma A.2.

The normal bundle of Gr⁢(2,V)⊂P⁢(⋀2V) satisfies

NGr⁢(2,V)/ℙ⁢(⋀2V)≅⋀2𝒬⁢(1)≅(⋀n-4𝒬∨)⁢(2),

where n=dim⁡V.

Proof.

The normal bundle fits into a commutative diagram

with exact rows and columns. Here, the map 𝒰∨⊗𝒰→𝒪 is given by evaluation. The map 𝒰∨⊗V→⋀2V⊗𝒪⁢(1) can be described as follows. Since det⁡(𝒰∨)≅𝒪⁢(1), there is a natural isomorphism 𝒰∨≅𝒰⁢(1), and the map in question is the composition

𝒰∨⊗V≅𝒰⊗V⁢(1)↪V⊗V⊗𝒪⁢(1)→⋀2V⊗𝒪⁢(1).

The sheaf ℰ is by definition the cokernel of this map. Due to the exact sequence

0→𝒰→V⊗𝒪→𝒬→0

we therefore have an isomorphism ℰ≅(⋀2𝒬)⁢(1). Hence also ℰ≅(⋀n-4𝒬∨)⁢(2) in view of the isomorphism det⁡(𝒬)≅𝒪⁢(1). It remains to note that ℰ≅NGr⁢(2,V)/ℙ⁢(⋀2V) by the snake lemma. ∎

A.2 Computations on Gr

From now on, we assume dim⁡V=5, fix an identification ⋀2V≅W, and let Gr⊂ℙ denote the corresponding embedded Grassmannian Gr⁢(2,V).

Lemma A.3.

The ideal sheaf IGr/P of Gr⊂P admits a resolution of the form

(A.1)0→𝒪⁢(-5)→V∨⊗𝒪⁢(-3)→V⊗𝒪⁢(-2)→ℐGr/ℙ→0.

Proof.

By regarding Gr⊂ℙ as a Pfaffian variety, this follows from [7] (see [14, Theorem 2.2] for a statement of the result in the form that we apply it). ∎

Lemma A.4.

The restriction map H0⁢(P,TP)→H0⁢(Gr,TP|Gr) is an isomorphism.

Proof.

Taking cohomology of the exact sequence

0→ℐGr/ℙ⊗Tℙ→Tℙ→Tℙ|Gr→0,

we see it is enough to show Hk⁢(ℙ,ℐGr/ℙ⊗Tℙ)=0 for k=0,1. In fact, we claim the sheaf ℐGr/ℙ⊗Tℙ has no cohomology. Indeed, R⁢Γ⁢(ℙ,Tℙ⁢(-t))≅0 for 2≤t≤9, as can be seen from the exact sequence

0→𝒪→W⊗𝒪⁢(1)→Tℙ→0,

so the claim follows by tensoring the resolution (A.1) with Tℙ and taking cohomology. ∎

Lemma A.5.

We have

R⁢Γ⁢(Gr,𝒬⁢(-t))≅{V⁢[0],t=0,0,1≤t≤5,Σ(t-2,t-2,t-3,3,3)⁢V∨⁢[-6],t≥6,

R⁢Γ⁢(Gr,⋀2𝒬⁢(-t))≅{⋀2V⁢[0],t=0,0,1≤t≤5,Σ(t-2,t-3,t-3,3,3)⁢V∨⁢[-6],t≥6.

Proof.

Note that 𝒬⁢(-t)≅Σ(t,t,t-1)⁢𝒬∨ and ⋀2𝒬⁢(-t)≅Σ(t,t-1,t-1)⁢𝒬∨. Now the result follows from Proposition A.1. ∎

Lemma A.6.

For 2≤t≤6, we have R⁢Γ⁢(Gr,NGr/P⁢(-t))≅0.

Proof.

Combine Lemmas A.2 and A.5. ∎

A.3 Computations on a GPK3 threefold

Let X=Gr1∩Gr2 be a GPK3 threefold. We write 𝒬i for the tautological rank 3 quotient bundle on Gri, and Ni=NGri/ℙ for the normal bundle of Gri⊂ℙ.

Lemma A.7.

For i=1,2, the ideal sheaf IX/Gri of X⊂Gri admits a resolution of the form

(A.2)0→𝒪⁢(-5)→V∨⊗𝒪⁢(-3)→V⊗𝒪⁢(-2)→ℐX/Gri→0.

Proof.

Analogously to Lemma A.3, this follows by regarding X⊂Gri as a Pfaffian variety. ∎

Lemma A.8.

The class of X in the Chow ring of Gri is 5⁢H3, where H denotes the Plücker hyperplane class.

Proof.

By (A.2) there is a resolution of 𝒪X on Gri of the form

(A.3)0→𝒪⁢(-5)→V∨⊗𝒪⁢(-3)→V⊗𝒪⁢(-2)→𝒪→𝒪X→0.

The result follows by taking ch3. ∎

Lemma A.9.

For t≥1 we have

H0⁢(X,𝒬i|X⁢(-t⁢H))=H0⁢(X,⋀2(𝒬i|X)⁢(-t⁢H))=0.

Proof.

From (A.3) we get a resolution

0→𝒬i⁢(-(t+5))→V∨⊗𝒬i⁢(-(t+3))→V⊗𝒬i⁢(-(t+2))

→𝒬i⁢(-t⁢H)→𝒬i|X⁢(-t⁢H)→0.

Let ℛi∙ be the complex concentrated in degrees [-3,0] given by the first four terms, so that there is a quasi-isomorphism

ℛi∙≃𝒬i|X⁢(-t⁢H).

Then the resulting spectral sequence

E1p,q=Hq⁢(X,ℛip)⇒Hp+q⁢(X,𝒬i|X⁢(-t⁢H))

combined with Lemma A.5 shows H0⁢(X,𝒬i|X⁢(-t⁢H))=0 for t≥1. The same argument also proves H0⁢(X,⋀2(𝒬i|X)⁢(-t⁢H))=0 for t≥1. ∎

Lemma A.10.

The restriction maps

V≅H0⁢(Gri,𝒬i)→H0⁢(X,𝒬i|X),i=1,2,

W∨≅H0⁢(ℙ,𝒪ℙ⁢(1))→H0⁢(X,𝒪X⁢(1))

are isomorphisms.

Proof.

Taking cohomology of the exact sequence

0→ℐX/Gri⊗𝒬i→𝒬i→𝒬i|X→0,

the first claim follows from the vanishing R⁢Γ⁢(Gri,ℐX/Gri⊗𝒬i)=0, which is a consequence of the resolution (A.2) combined with Lemma A.5. The second claim is proved similarly. ∎

Lemma A.11.

The restriction maps H0⁢(Gri,Ni)→H0⁢(X,Ni|X), i=1,2, are isomorphisms.

Proof.

Taking cohomology of the exact sequence

0→ℐX/Gri⊗Ni→Ni→Ni|X→0,

we see it is enough to show Hk⁢(Gri,ℐX/Gri⊗Ni)=0 for k=0,1. In fact, we claim the sheaf ℐX/Gri⊗Ni has no cohomology. This follows by tensoring the resolution (A.2) with Ni and applying Lemma A.6. ∎

Acknowledgements

We are grateful to Johan de Jong for very useful conversations about this work. We also benefited from discussions with Ron Donagi, Sasha Kuznetsov, and Daniel Litt. We thank Michał Kapustka for interesting comments and for informing us about the history of GPK3 threefolds. We thank John Ottem and Jørgen Rennemo for coordinating the release of their paper with ours.

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Received: 2017-07-18

Revised: 2018-04-12

Published Online: 2018-05-29

Published in Print: 2020-03-01

Intersections of two Grassmannians in ℙ⁹

Abstract

A Borel–Weil–Bott computations

A.1 Borel–Weil–Bott

Proposition A.1.

Lemma A.2.

Proof.

A.2 Computations on Gr

Lemma A.3.

Proof.

Lemma A.4.

Proof.

Lemma A.5.

Proof.

Lemma A.6.

Proof.

A.3 Computations on a GPK3 threefold

Lemma A.7.

Proof.

Lemma A.8.

Proof.

Lemma A.9.

Proof.

Lemma A.10.

Proof.

Lemma A.11.

Proof.

Acknowledgements

References

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