# Hyperbolicity of varieties supporting a variation of Hodge structure

@article{Brunebarbe2017HyperbolicityOV,
title={Hyperbolicity of varieties supporting a variation of Hodge structure},
journal={arXiv: Algebraic Geometry},
year={2017}
}
• Published 5 July 2017
• Mathematics
• arXiv: Algebraic Geometry
We generalize former results of Zuo and the first author showing some hyperbolicity properties of varieties supporting a variation of Hodge structure. Our proof only uses the special curvature properties of period domains. In particular, in contrast to the former approaches, it does not use any result on the asymptotic behaviour of the Hodge metric.
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## References

SHOWING 1-10 OF 26 REFERENCES

### Symmetric differentials and variations of Hodge structures

• Yohan Brunebarbe
• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2018
Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures

### Singular Hermitian metrics and positivity of direct images of pluricanonical bundles

• Mihai Păun
• Mathematics
Algebraic Geometry: Salt Lake City 2015
• 2018
This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for

### On the negativity of kernels of Kodaira–Spencer maps on Hodge bundles and applications

Viehweg recently asked that if the cotangent bundles of moduli varieties of polarized varieties with log-pole along infinity are positive in some sense. It is well known that the moduli space of

### Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

• Mathematics
• 2013
In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to

### ON THE VOLUME OF A LINE BUNDLE

Using the Calabi–Yau technique to solve Monge-Ampere equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic

### Symmetric differentials and the fundamental group

• Mathematics
• 2012
Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a

### Positivity of twisted relative pluricanonical bundles and their direct images

• Mathematics
• 2014
Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian

### Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995. The main goal of the notes is to study complex varieties

### Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces

Let V and W be non-singular projective varieties over the field of complex numbers C, n= dim (V) and m=dim (W). Let/: V---+W be a fibre space (this simply means that I is surjective with connected

### Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun

• Mathematics
• 2016
We present a simplified proof for a recent theorem by Junyan Cao and Mihai Paun, confirming a special case of Iitaka's conjecture: if $f \colon X\to Y$ is an algebraic fiber space, and if the