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Yohan Brunebarbe

Publications and source records attributed to Yohan Brunebarbe.

16 recordsLinked to original sources
arXiv2024

The linear Shafarevich conjecture for quasiprojective varieties and algebraicity of Shafarevich morphisms

We prove that the universal cover of a normal complex algebraic variety admitting a faithful complex representation of its fundamental group is an analytic Zariski open subset of a holomorphically convex complex space. This is a non-proper version of the Shafarevich conjecture. More generally we define a class of subset of the Betti stack for which the covering space trivializing the corresponding local systems has this property. Secondly, we show that for any complex local system $V$ on a normal complex algebraic variety $X$ there is an algebraic map $f \colon X\to Y$ contracting precisely the subvarieties on which $V$ is isotrivial.

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arXiv2023

Existence of the Shafarevich morphism for semisimple local systems on quasi-projective varieties

Let X be a normal connected complex algebraic variety equipped with a semisimple complex representation of its fundamental group. Then, under a maximality assumption, we prove that the covering space of X associated to the kernel of the representation has a proper surjective holomorphic map with connected fibres onto a normal analytic space with no positive-dimensional compact analytic subspace.

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arXiv2022

o-minimal GAGA and a conjecture of Griffiths

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.

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arXiv2022

An algebraic approach to the hyperbolicity of moduli stacks of Calabi-Yau varieties

The moduli stacks of Calabi-Yau varieties are known to enjoy several hyperbolicity properties. The best results have so far been proven using sophisticated analytic tools such as complex Hodge theory. Although the situation is very different in positive characteristic (e.g. the moduli stack of principally polarized abelian varieties of dimension at least 2 contains rational curves), we explain in this note how one can prove many hyperbolicity results by reduction to positive characteristic, relying ultimately on the nonabelian Hodge theory in positive characteristic developed by Ogus and Vologodsky.

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arXiv2022

Counting integral points of bounded height on varieties with large fundamental group

The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. Such an improvement, i.e. requiring large fundamental group as opposed to the existence of a geometric variation of pure Hodge structures, was already asked in op.cit..

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arXiv2020

Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications

We prove a Second Main Theorem type inequality for any log-smooth projective pair $(X,D)$ such that $X\setminus D$ supports a complex polarized variation of Hodge structures. This can be viewed as a Nevanlinna theoretic analogue of the Arakelov inequalities for variations of Hodge structures due to Deligne, Peters and Jost-Zuo. As an application, we obtain in this context a criterion of hyperbolicity that we use to derive a vast generalization of a well-known hyperbolicity result of Nadel. The first ingredient of our proof is a Second Main Theorem type inequality for any log-smooth projective pair $(X,D)$ such that $X\setminus D$ supports a metric whose holomorphic sectional curvature is bounded from above by a negative constant. The second ingredient of our proof is an explicit bound on the holomorphic sectional curvature of the Griffiths-Schmid metric constructed from a variation of Hodge structures. As a byproduct of our approach, we also establish a Second Main Theorem type inequality for pairs $(X,D)$ such that $X\setminus D$ is hyperbolically embedded in $X$.

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arXiv2020

Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures

Looking at the finite étale congruence covers $X(p)$ of a complex algebraic variety $X$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, we show that both the minimal gonality among all curves contained in $X(p)$ and the minimal volume among all subvarieties of $X(p)$ tend to infinity with $p$. This applies for example to Shimura varieties, moduli spaces of curves, moduli spaces of abelian varieties, moduli spaces of Calabi-Yau varieties, and can be made effective in many cases. The proof goes roughly as follows. We first prove a generalization of the Arakelov inequalities valid for any variation of Hodge structures on higher-dimensional algebraic varieties, which implies that the hyperbolicity of the subvarieties of $X$ is controlled by the positivity of a single line bundle. We then show in general that a big line bundle on a normal proper algebraic variety $\bar X$ can be made more and more positive by going to finite covers of $\bar X$ defined using level structures of a local system defined on a Zariski-dense open subset.

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arXiv2020

Quasiprojectivity of images of mixed period maps

We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasiprojective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions. Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating a $\mathbb{R}_{an,exp}$-definable structure to mixed period domains and admissible mixed period maps.

math.AG↗
arXiv2020

Definability of mixed period maps

We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}_{alg}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in $\mathbb{R}_{an,exp}$ with respect to this structure. As a consequence we reprove that the zero loci of admissible normal functions are algebraic.

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arXiv2017

Semi-positivity from Higgs bundles

We prove a generalization of the Fujita-Kawamata-Zuo semi-positivity Theorem for filtered regular meromorphic Higgs bundles and tame harmonic bundles. Our approach gives a new proof in the cases already considered by these authors. We give also an application to the geometry of smooth quasi-projective complex varieties admitting a semisimple complex local system with infinite monodromy.

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arXiv2017

Hyperbolicity of varieties supporting a variation of Hodge structure

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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arXiv2016

A strong hyperbolicity property of locally symmetric varieties

We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. This result corresponds through the Green-Griffiths-Lang's conjecture to a well-known result of Nadel.

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arXiv2015

Fundamental group and pluridifferentials on compact Kähler manifolds

A compact Kähler manifold is shown to be simply-connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map between connected complex manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if the domain is Kähler.

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arXiv2013

Symmetric differentials and the fundamental group

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 sense, this would mean that every variety with infinite fundamental group has some nonpositive curvature. We show that the answer to Esnault's question is positive when the fundamental group has a finite-dimensional representation over some field with infinite image. This applies to all known varieties with infinite fundamental group. Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures. One interest of these results is that symmetric differentials give information in the direction of Kobayashi hyperbolicity. For example, they limit how many rational curves the variety can contain.

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