
Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures. To appear in L'Épijournal de Géométrie Algébrique. arXiv:2001.04426
Abstract Oberwolfach report
In this paper we study various hyperbolicity properties for a quasicompact Kähler manifold \(U\) which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero dimensional. In the first part we prove that \(U\) is algebraically hyperbolic, and that the generalized big Picard theorem holds for \(U\). In the second part, we prove that there is a finite étale cover \(\tilde{U}\) of \(U\) from a quasiprojective manifold \(\tilde{U}\) such that any projective compactification \(X\) of \(\tilde{U}\) is Picard hyperbolic modulo the boundary \(X\tilde{U}\), and any irreducible subvariety of \(X\) not contained in \(X\tilde{U}\) is of general type.
This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion free lattices.

A characterization of complex quasiprojective manifolds uniformized by unit balls. (with an appendix written jointly with Benoît Cadorel). To appear in Math. Ann. arXiv:2006.16178
Abstract
In 1988 Simpson extended the DonaldsonUhlenbeckYau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasiprojective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasiprojective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson.
Several byproducts are also obtained in this paper.
 On the hyperbolicity of base spaces for maximal variational families of smooth projective varieties. (with an appendix by Dan Abramovich). arXiv:1806.01666 To appear in Journal of the European Mathematical Society (JEMS). Abstract
For maximal variational smooth families of projective manifolds whose general fibers have semiample canonical bundle, the Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by CampanaPaun and later generalized by PopaSchnell. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasiprojective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semiample canonical bundle. This proves a conjecture by ViehwegZuo in 2003. We also prove the Kobayashi hyperbolicity of base spaces of effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by ToYeung, in which they further assumed that these families are canonically polarized.
 Kobayashi hyperbolicity of the complements of general hypersurfaces of high degrees. (joint work with Damian Brotbek ), arXiv:1804.01719. Geometric And Functional Analysis (GAFA), June 2019, Volume 29, Issue 3, pp 690–750. link Abstract
In this paper, we prove that in any projective manifold, the
complements of general hypersurfaces of sufficiently large degree are
Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.
 On the DiverioTrapani Conjecture. arXiv:1703.07560 . Ann. Scient. Éc. Norm. Sup. 4 e série, t. 53, 2020, p. 787 à 814. link Abstract
In this paper we establish effective lower bounds on the degrees of the Debarre and Kobayashi conjectures. Then we study a more general conjecture proposed by DiverioTrapani on the ampleness of jet bundles of general complete intersections in complex projective spaces.
 On the positivity of the logarithmic cotangent bundle. (joint work with Damian Brotbek ) arXiv:1712.09887. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, p. 30013051 (en l'honneur du professeur JeanPierre Demailly). link Abstract
The aim of this work is to construct examples of pairs whose
logarithmic cotangent bundles have strong positivity properties. These
examples are constructed from any smooth ndimensional complex
projective varieties by considering the sum of at least n general
sufficiently ample hypersurfaces. This result can be seen as a
logarithmic counterpart of the Debarre conjecture, proven by
BrotbekDarondeau link and Xie independently.
 Applications of the OhsawaTakegoshi
Extension Theorem to Direct Image
Problems. Int. Math. Res. Not. IMRN, rnaa018
Abstract
In the first part of the paper, we study a Fujitatype conjecture by Popa and Schnell, and give an effective bound on the generic global generation of the direct image of the twisted pluricanonical bundle. We also point out the relation between the Seshadri constant and the optimal bound. In the second part, we prove a conjecture by DemaillyPeternellSchneider in a more general setting. As a byproduct, we generalize the theorems by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal cases, and refine a result by Broustet and Pacienza on the rational connectedness of the image.
 Transcendental morse inequality and generalized Okounkov bodies algebraic geometry. Algebraic Geometry 4 (2) (2017) 177–202. Link Abstract
The main goal of this article is to construct "generalized Okounkov
bodies" for an arbitrary transcendental pseudoeffective (1,1)class on
a Kähler manifold. We give a complete characterization of generalized Okounkov bodies on surfaces, and relate the standard Euclidean
volume of the body to the volume of the corresponding big class as
defined by Boucksom; this solves a problem raised by Lazarsfeld and
Musţată in the case of surfaces.

Simpson correspondence for
semistable Higgs bundles over Kähler
manifolds. hal02391629 . Pure and Applied Mathematics Quarterly Volume 17, Number 5, 18991911, 2021. Link Abstract
In this note we provide an elementary proof of the Simpson
correspondence between semistable Higgs bundles with vanishing Chern
classes and representation of fundamental groups of Kähler manifolds.
 Kobayashi measure hyperbolicity for singular directed varieties of general type. Comptes Rendus Mathématique,
Volume 354, Issue 9, (2016), Pages 920924.
Link Abstract
In this note, we prove the nondegeneracy of the KobayashiEisenman volume measure of
the singular directed varieties in the sense of Demailly.
