Positivity in Kähler–Einstein theory

Cerbo, Gabriele Di; Cerbo, Luca F. Di

doi:10.1017/s0305004115000377

Cited by 7 publications

(14 citation statements)

References 38 publications

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“…We then conclude that (K X + D) · C i ≥ 2 for all i. Following the proof of Theorem 2.1 in [DiC12] and using the bound given by Theorem 2.3, it follows that the K X + αD is ample for all α ∈ 1 3 , 1 .…”

Section: A Gap Theoremmentioning

confidence: 56%

“…Because of Theorem 1.1, for all α close to one, we know K X + αD is ample. By Corollary 4.18 in [DiC12], it follows that K X + D is strictly nef. On the other hand, we must…”

Section: A Gap Theoremmentioning

confidence: 86%

“…The proof of Theorem 1.1 is based on the techniques developed by the authors in [DiC12]. We list here some of the main applications.…”

Section: Introductionmentioning

confidence: 99%

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Effective results for complex hyperbolic manifolds

Cerbo

Cerbo²

2014

Journal of the London Mathematical Society

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Abstract. The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We estimate the number of ends of such manifolds in terms of their volume. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we give two sided bounds on their Picard numbers in terms of the volume and the number of cusps.

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Section: A Gap Theoremmentioning

confidence: 56%

“…Because of Theorem 1.1, for all α close to one, we know K X + αD is ample. By Corollary 4.18 in [DiC12], it follows that K X + D is strictly nef. On the other hand, we must…”

Section: A Gap Theoremmentioning

confidence: 86%

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Effective results for complex hyperbolic manifolds

Cerbo

Cerbo²

2014

Journal of the London Mathematical Society

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“…A natural question is whether there exist uniform bounds on the asymptotic range of β; and if so, what do they depend on? This was first addressed by Di Cerbo-Di Cerbo [86] in the case β = β 1 (1, . .…”

Section: Uniform Boundsmentioning

confidence: 99%

Smooth and Singular Kähler–Einstein Metrics

Rubinstein¹

2014

Geometric and Spectral Analysis

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Smooth Kähler-Einstein metrics have been studied for the past 80 years. More recently, singular Kähler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a survey of some of these developments.The Kähler-Einstein (KE) equation is among the oldest fully nonlinear equations in modern geometry. A wide array of tools have been developed or applied towards its understanding, ranging from Riemannian geometry, PDE, pluripotential theory, several complex variables, microlocal analysis, algebraic geometry, probability, convex analysis, and more. The interested reader is referred to the numerous existing surveys on related topics

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“…Recall that a divisor class is nef modulo the boundary if it intersects positively with any integral curve not contained in the boundary (see for example Section 5 of [DCDC12]). We thus conclude the Corollary 2.11.…”

Section: Curves In Quotients Of H Nmentioning

confidence: 99%

The geometric torsion conjecture for abelian varieties with real multiplication

Bakker

Tsimerman

2018

J. Differential Geom.

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The geometric torsion conjecture asserts that the torsion part of the Mordell-Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we furthermore show that the torsion is bounded in terms of the gonality of the base curve, which is the closer analog of the arithmetic conjecture. The proof is a hybrid technique employing both the hyperbolic and algebraic geometry of the toroidal compactifications of the Hilbert modular varieties X(1) parametrizing such abelian varieties. We show that only finitely many torsion covers X1(n) contain d-gonal curves outside of the boundary for any fixed d. We further show the same is true for entire curves C → X1(n). Statement of ResultsFor any elliptic curve E/Q, the group of rational points E(Q) is finitely generated by Mordell's theorem. The free part behaves wildly; it is expected that there are elliptic curves E/Q with arbitrarily large rank rk E(Q), and the record to date is an elliptic curve E with rk E(Q) ≥ 28 found by Elkies. On the other hand, by a celebrated theorem of Mazur [MG78] the torsion part E(Q) tor is uniformly bounded:Theorem (Mazur). For any elliptic curve E/Q, |E(Q) tor | ≤ 16.Mazur's theorem was subsequently generalized to arbitrary number fields K/Q by Merel [Mer96] (building on partial results of [Kam92]) who showed a stronger uniformity: there is an integer N = N (d) such that for any degree d number field K and any elliptic curve E/K, every K-rational torsion point has order dividing N , i.e. E(K) tor ⊂ E(K) [N ].Similarly, it is expected that the torsion part of the Mordell-Weil group of an abelian variety A/K is uniformly bounded, though there are few results in this direction. The same question can be asked for K = k(C) the function field of a curve C over any field k, and though k = F p is most closely analogous to the number field case, k = Q is also interesting:Conjecture (Geometric torsion conjecture). Let k be an algebraically closed field of characteristic 0. There is an integer N = N (g, n) such that for any Date: April 9, 2015.

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Positivity in Kähler–Einstein theory

Cited by 7 publications

References 38 publications

Effective results for complex hyperbolic manifolds

Effective results for complex hyperbolic manifolds

Smooth and Singular Kähler–Einstein Metrics

The geometric torsion conjecture for abelian varieties with real multiplication

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