Effective results for complex hyperbolic manifolds

Cerbo, Gabriele Di; Cerbo, Luca F. Di

doi:10.1112/jlms/jdu065

Cited by 13 publications

(25 citation statements)

References 33 publications

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“…A similar argument is used by Di Cerbo-Di Cerbo to give an improvement to Parker's cusp bound in dimensions 2 [DCDC14] and 3 [DCDC15a]. Note that by (1) we have vol(X) = (4π) n n!…”

Section: Ampleness and Applicationsmentioning

confidence: 69%

“…On the other hand, we already know K X + (1 − ǫ)D is ample for small ǫ > 0 (cf. [DCDC15a]). The interior of any line drawn between a point of the nef cone and a point in the ample cone is contained in the ample cone, and K X + tD for t ∈ [1 − n+1 2π , 1 − ǫ] is such a line.…”

Section: Ampleness and Applicationsmentioning

confidence: 99%

“…By (1) and Proposition 3.3, the divisor is nef modulo the boundary, but for any component of the boundary is ample so it is in fact nef. Moreover, is ample for all sufficiently small (see [DiCDiC15]). As the interior of any line drawn between a point of the nef cone and a point in the ample cone is contained in the ample cone, the claim follows.◻…”

Section: Boundary Multiplicity Inequalities and Amplenessmentioning

confidence: 99%

“…These methods use as input the positivity of divisors of the form ; for it comes for free on any toroidal compactification. Di Cerbo and Di Cerbo [DiCDiC15] have systematically studied effectivity results that follow from this positivity in the range (or more recently for in [DiCDiC]), including bounds on the number, degree, and Picard rank of hyperbolic manifolds of a given volume. For most of these results, simply plugging Theorem A into their argument yields a better bound, and we choose to leave these modifications to the reader.…”

Section: Introductionmentioning

confidence: 99%

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The Kodaira dimension of complex hyperbolic manifolds with cusps

Bakker¹,

Tsimerman²

2017

Compositio Math.

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We prove a bound relating the volume of a curve near a cusp in a complex ball quotient X = B/Γ to its multiplicity at the cusp. There are a number of consequences: we show that for an n-dimensional toroidal compactification X with boundary D, K X +(1−λ)D is ample for λ ∈ (0, (n + 1)/2π), and in particular that K X is ample for n 6. By an independent algebraic argument, we prove that every ball quotient of dimension n 4 is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green-Griffiths conjecture.

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“…A similar argument is used by Di Cerbo-Di Cerbo to give an improvement to Parker's cusp bound in dimensions 2 [DCDC14] and 3 [DCDC15a]. Note that by (1) we have vol(X) = (4π) n n!…”

Section: Ampleness and Applicationsmentioning

confidence: 69%

Section: Ampleness and Applicationsmentioning

confidence: 99%

Section: Boundary Multiplicity Inequalities and Amplenessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Kodaira dimension of complex hyperbolic manifolds with cusps

Bakker¹,

Tsimerman²

2017

Compositio Math.

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“…where NE(X) K X ≥0 are the points in NE(X) that pair nonnegatively with K X , and where the {C i } are the (possibly countably many) extremal rays. See [DD15,§2] for further details and references.…”

Section: Totally Geodesic (And Other) Punctured Spheresmentioning

confidence: 99%

Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications

Cerbo

Stover

2019

Trans. Amer. Math. Soc.

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In this paper, we study punctured spheres in two dimensional ball quotient compactifications (X, D). For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of K X + αD for α ∈ ( 1 4 , 1), giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.

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On effective existence of symmetric differentials of complex hyperbolic space forms

Wong

2018

Math. Z.

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For a noncompact complex hyperbolic space form of finite volume X = B n /Γ, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification X of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity X − X called canonical radius of a cusp of Γ. The main result in the article is that there is a constant r * = r * (n) depending only on the dimension, so that if the canonical radii of all cusps of Γ are larger than r * , then there exist symmetric differentials of X vanishing at infinity. As a corollary, we show that the cotangent bundle T X is ample modulo the infinity if moreover the injectivity radius in the interior of X is larger than some constant d * = d * (n) which depends only on the dimension.

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Effective results for complex hyperbolic manifolds

Cited by 13 publications

References 33 publications

The Kodaira dimension of complex hyperbolic manifolds with cusps

The Kodaira dimension of complex hyperbolic manifolds with cusps

Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications

On effective existence of symmetric differentials of complex hyperbolic space forms

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