Boundedness of elliptic Calabi-Yau varieties with a rational section

Birkar, Caucher; Cerbo, Gabriele Di; Svaldi, Roberto

doi:10.48550/arxiv.2010.09769

Cited by 4 publications

(13 citation statements)

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“…All eigenvalues of g thus have order 2 or 6, so g has at least three eigenvalues of order 6. As g has determinant 1, we only have two possibilities: There is a generator of g similar to diag(−1, ω, ω, ω), or diag(ω, ω 5 , ω, ω 5 ).…”

Section: C Group Theoretical Treatment Of a Point's Stabilizer In Dim...mentioning

confidence: 99%

“…Subsection 4.B exhibits a correlation between the type of g and the isogeny type (possibly even isomorphism type) of the abelian variety B on which it acts. A corollary is that if g, h ∈ PStab(W ) are two junior elements, then they should either have the same type, or one is of type (1 n−4 , ω, ω, ω, −1) and the other (1 n−3 , j, j, j), or one is of type (1 n−4 , i, i, i, i) and the other 5 12 , ζ 5 12 . In particular, if PStab(W ) is cyclic, it must indeed be generated by one junior element.…”

Section: }}mentioning

confidence: 99%

“…For (9), assume that h is a junior element of order 3. By (5), it fixes a point τ ∈ B, and a translated abelian subvariety W ′ + τ , where W ′ is an abelian subvariety of codimension 3 in…”

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: C Group Theoretical Treatment Of a Point's Stabilizer In Dim...mentioning

confidence: 99%

Section: }}mentioning

confidence: 99%

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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“…Similarly, the results in [Li20] apply to the study of fibrations of Fano type. The boundedness of varieties with an elliptic fibration is considered in [Bir18], [BDCS20], [CDCH + 18] and [Fil20]. By analogy with the definition of volumes of divisors, the Iitaka volume of a Q-divisor is defined as follows.…”

Section: Introductionmentioning

confidence: 99%

Boundedness of polarised Calabi-Yau fibrations

Jiao¹

2022

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Singularities on toric fibrations

Birkar¹,

Chen²

2021

Sb. Math.

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In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M Kernan which roughly says that if is an -lc Fano-type log Calabi-Yau fibration, then the singularities of the log base are bounded in terms of and where and are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if is a toric Fano fibration with being -lc, then the multiplicities of the fibres over codimension one points are bounded depending only on and . Bibliography: 20 titles.

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Boundedness of elliptic Calabi-Yau varieties with a rational section

Cited by 4 publications

References 20 publications

Finite quotients of abelian varieties with a Calabi-Yau resolution

Finite quotients of abelian varieties with a Calabi-Yau resolution

Boundedness of polarised Calabi-Yau fibrations

Singularities on toric fibrations

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