Higher dimensional Calabi–Yau manifolds of Kummer type

Burek, Dominik

doi:10.1002/mana.201800487

Cited by 7 publications

(9 citation statements)

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“…• Z 7 k , then there are k generators of it such that their matrices are similar to diag(1 n−3 , ζ 7 , ζ 7 2 , ζ 7 4 ), and all eigenspaces of these matrices with eigenvalues other than 1 are in direct sum.…”

Section: Theorem 15 Let a Be An Abelian Variety Of Dimension N And G ...mentioning

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: Theorem 15 Let a Be An Abelian Variety Of Dimension N And G ...mentioning

confidence: 99%

“…Similar examples of dimension 3 are numerous, as in [32,31,30], and even less well understood in higher dimensions, cf. [9,10,35,2,7]. In arbitrary dimension, it is known that a crepant resolution or terminalization only changes the type of a klt K-trivial variety if its decomposition entails an abelian factor ( [13,Prop.4.10]).…”

Section: Introductionmentioning

confidence: 99%

Finite quotients of abelian varieties with a Calabi-Yau resolution

Gachet¹

2022

Preprint

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“…[43,Section 4.3.5.2]. Conjecture 4.1 has been proven in some scattered special cases [41], [24], [25], [26], [27], [6], [29], [7], [28], but is still wide open for a general K3 surface.…”

Section: Robert Laterveermentioning

confidence: 99%

On the Chow ring of some special Calabi–Yau varieties

Laterveer

2021

Can. Math. Bull.

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We consider Calabi-Yau n-folds X arising from certain hyperplane arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of X generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin's conjecture for X, and Voevodsky's smash-nilpotence conjecture for odd-dimensional X.

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“…The aim of this work is to give a formula for the Hodge numbers and zeta function of Kum 3 (E, G) using the Chen-Ruan orbifold cohomology theory ( [CR04]) and the description of the Frobenius action on the orbifold cohomology ( [Ros07]). In [Bur18], [Bur20], [Bur21] we successfully used that approach in order to compute Hodge numbers and zeta function of manifolds (X 1 × X 2 × . .…”

Section: Introductionmentioning

confidence: 99%