Birational boundedness of rationally connected Calabi–Yau 3-folds

Chen, Weichung; Cerbo, Gabriele Di; Han, Jingjun; Jiang, Chen; Svaldi, Roberto

doi:10.1016/j.aim.2020.107541

Cited by 21 publications

(18 citation statements)

References 29 publications

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“…Up to exchanging g and h, we can assume k ∈ g . If ρ(k) is similar to diag(1, j, j, j) or diag(1, j 2 , j 2 , j 2 ), then respectively ρ(gk) or ρ(g 2 k) has no 1 as an eigenvalue, which contradicts (8). Else, ρ(k) is similar to diag(1, 1, j, j 2 ).…”

Section: Proof Of Theorem 14mentioning

confidence: 97%

“…For (8), assume that h fixes a point τ in A and has even order. For some β, h β has order 2, and thus equals g α : So, every fixed point of h is a fixed point of g α .…”

Section: Regarding (4) We Note That For H ∈ G W H(b) = M (H)b + T (H...mentioning

confidence: 99%

“…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. These three main families of K-trivial varieties are the subject of large, mostly disjoint realms of the literature, ranging from the well-known theory of abelian varieties (exposed notably in the reference books [6,39]), through the thriving study of hyperkähler varieties (see [11,1,19] for surveys), to the unruly "Zoo of Calabi-Yau varieties", populated by a huge amount of examples ( [24,25] for K3 surfaces and Calabi-Yau threefolds embedded as hypersurfaces in toric varieties only), and whose boundedness is yet not established (see [44,45,8,12,5] for recent breakthroughs).…”

Section: Introductionmentioning

confidence: 99%

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Finite quotients of abelian varieties with a Calabi-Yau resolution

Gachet¹

2022

Preprint

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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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Section: Proof Of Theorem 14mentioning

confidence: 97%

“…For (8), assume that h fixes a point τ in A and has even order. For some β, h β has order 2, and thus equals g α : So, every fixed point of h is a fixed point of g α .…”

Section: Regarding (4) We Note That For H ∈ G W H(b) = M (H)b + T (H...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finite quotients of abelian varieties with a Calabi-Yau resolution

Gachet¹

2022

Preprint

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“…As we saw in the Introduction, log birational boundedness of Calabi–Yau pairs does not hold if we allow ourselves also to consider product-type pairs. Nonetheless, it is possible that the boundedness can still be proven if we consider -dimensional product-type pairs with rationally connected and the coefficients of vary in a fixed DCC set; see [CDHJS21, Conjecture 1.3]. A proof of this fact appears in [BDS20, Theorem 1.4].…”

Section: A Structure Theorem For Calabi–yau Pairsmentioning

confidence: 99%

Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section

Cerbo¹,

Svaldi²

2021

Compositio Math.

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We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim (Y)\leq 5$ and $Y$ is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs $(X, \Delta )$ with $K_X+\Delta$ numerically trivial and not of product type, in dimension at most four.

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“…Some recent results were also obtained in the case of K-trivial varieties, cf. [11,8,6]. In between varieties of general type and K-trivial ones, we have varieties of intermediate Kodaira dimension.…”

Section: The Canonical Bundle Formula and Generalized Pairs Varieties Of Intermediate Kodaira Dimensionmentioning

confidence: 99%

Invariance of Plurigenera and boundedness for Generalized Pairs

Filipazzi¹,

Svaldi²

2020

RMC

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In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension κ(X) = dim(X) − 1.

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Birational boundedness of rationally connected Calabi–Yau 3-folds

Cited by 21 publications

References 29 publications

Finite quotients of abelian varieties with a Calabi-Yau resolution

Finite quotients of abelian varieties with a Calabi-Yau resolution

Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section

Invariance of Plurigenera and boundedness for Generalized Pairs

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