Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties

Bakker, Benjamin; Guenancia, Henri; Lehn, Christian

doi:10.48550/arxiv.2012.00441

Cited by 5 publications

(13 citation statements)

References 23 publications

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“…As a consequence of the results of [BGL20], a locally trivial deformation of an irreducible symplectic variety is irreducible symplectic. We sketch the argument.…”

Section: Preliminariesmentioning

confidence: 89%

Deformations of rational curves on primitive symplectic varieties and applications

Lehn¹,

Mongardi²,

Pacienza³

2021

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We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective K3 or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.

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“…As a consequence of the results of [BGL20], a locally trivial deformation of an irreducible symplectic variety is irreducible symplectic. We sketch the argument.…”

Section: Preliminariesmentioning

confidence: 89%

Deformations of rational curves on primitive symplectic varieties and applications

Lehn¹,

Mongardi²,

Pacienza³

2021

Preprint

Self Cite

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“…As such, Λ is finitely-generated and torsion-free. But by [27], since k ∈ [ [3,20]] ∪ {24}, the ring of cyclotomic integers Z[ζ k ] is a principal ideal domain. So, by the structure theorem for finitely-generated modules over principal ideal domains,…”

Section: B the Abelian Varieties Corresponding To The Twelve Juniorsmentioning

confidence: 99%

“…Since singularities are a byproduct of the Minimal Model Program, studying singular varieties with trivial canonical class, or singular K-trivial varieties, is an important question in the birational classification of complex algebraic varieties. From this point of view, the recent generalization of the Beauville-Bogomolov decomposition theorem for smooth K-trivial varieties ( [4]) to klt K-trivial varieties ( [14,13,16,3]) is highly relevant. It indeed establishes that, after a finite quasiétale cover, any klt K-trivial variety is a product of a smooth abelian variety, some irreducible holomorphic symplectic varieties with canonical singularities, also called hyperkähler varieties, and some Calabi-Yau varieties with canonical singularities.…”

Section: Introductionmentioning

confidence: 99%

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Finite quotients of abelian varieties with a Calabi-Yau resolution

Gachet¹

2022

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Let A be an abelian variety, and G ⊂ Aut(A) a finite group acting freely in codimension two. We discuss whether the singular quotient A/G admits a resolution that is a Calabi-Yau manifold. While Oguiso constructed two examples in dimension 3 [32], we show that there are none in dimension 4. We also classify up to isogeny the possible abelian varieties A in arbitrary dimension.Remark 2.3. Note that if g ∈ Aut(A) admits a fixed point, then g contains no translation, so g and its matrix M (g) have the same order.Junior elements play a key role in the study of finite quotient singularities, as the following theorem emphasizes.Theorem 2.4. [18] Let C n /G be a finite Gorenstein quotient singularity, and suppose that it has a minimal model X. Then there is a natural one-to-one correspondence between conjugacy classes of junior elements in G and prime exceptional divisors in X.Remark 2.5. Note that such a minimal model X always exists as a relative minimal model of a resolution X → C n /G, by [22, 1.30.6].Quotient singularities are Q-factorial, so they can not be resolved by small birational morphisms. This yields a simple corollary of the theorem.Corollary 2.6. [18] Let C n /G be a finite Gorenstein quotient singularity, with G acting freely in codimension 1. If the singularity C n /G admits a crepant resolution, then there is a junior element g ∈ G.In fact, [18, Par.4.5] conjectures that under the same hypotheses, if the singularity C n /G admits a crepant resolution, then any maximal cyclic subgroup of G contains a junior element. A counterexample to this conjecture is however presented in Remark 8.15. In this section, we prove a weak version of that conjecture. We phrase it in an analytic set-up, as our later applications call for that, but the proof works in the affine set-up just as well.Proposition 2.7. Let G ⊂ GL n (C) be a finite group acting freely in codimension 1 on C n , and let U ⊂ C n be a G-stable simply-connected analytic neighborhood of 0 ∈ C n . If the singularity U/G admits a crepant resolution X G , then the group G is generated by junior elements.Note that a singularity admitting a crepant resolution is Gorenstein. By [20][42], the existence of a crepant resolution X G thus implies that G ⊂ SL n (C).In order to prove the proposition, we need some background in valuation theory.

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“…Generalizing the celebrated Beauville-Bogomolov decomposition, it was recently proved in [BGL20] that compact Kähler spaces X with klt singularities such that K X is numerically trivial should admit a finite cover X ′ → X, unramified in codimension one and such that X ′ = T × ∏ Y i × ∏ Z j where T is a (smooth) torus, Y i are irreducible Calabi-Yau varieties and Z j are irreducible holomorphic symplectic varieties, cf. also [Dru18, GGK19, HP19, CGGN20] for related anterior results.…”

Section: Hodge Classes Vs Transcendental Classesmentioning

confidence: 99%

Continuity of singular Kähler-Einstein potentials

Guedj¹,

Guenancia²,

Zériahi³

2020

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In this note, we investigate some regularity aspects for solutions of degenerate complex Monge-Ampère equations (DCMAE) on singular spaces.First, we study the Dirichlet problem for DCMAE on singular Stein spaces, showing a general continuity result. A consequence of our results is that Kähler-Einstein potentials are continuous at isolated singularities.Next, we establish the global continuity of solutions to DCMAE when the reference class belongs to the real Néron-Severi group. This yields in particular the continuity of Kähler-Einstein potentials on any irreducible Calabi-Yau variety. We show that the associated singular Kähler-Einstein metrics are strictly positive when the variety is moreover smoothable.

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Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties

Cited by 5 publications

References 23 publications

Deformations of rational curves on primitive symplectic varieties and applications

Deformations of rational curves on primitive symplectic varieties and applications

Finite quotients of abelian varieties with a Calabi-Yau resolution

Continuity of singular Kähler-Einstein potentials

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