Yohan Brunebarbe’s research while affiliated with Université Bordeaux-I and other places

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.

The main result of this paper 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. This generalizes a recent paper of Ellenberg, Lawrence and Venkatesh and replies to two questions asked therein.

We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, 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 an ℝ an , exp {\mathbb{R}_{\mathrm{an},\exp}} -definable structure to mixed period domains and admissible mixed period maps.

We equip integral graded-polarized mixed period spaces with a natural \mathbb{R}_{\mathrm{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}_{\mathrm{an,exp}} with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic.

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.

We propose a generalization of the Green-Griffiths-Lang conjecture to the relative setting and prove that a strong form of it holds for families of varieties of maximal Albanese dimension. A key step of the proof consists in a truncated second main theorem type estimate in Nevanlinna theory for families of abelian varieties.

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.

We prove that the projective complex algebraic varieties admitting a large complex local system satisfy a strong version of the Green-Griffiths-Lang conjecture.

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..

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.