
Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures. arXiv:2001.04426
Abstract
In this paper we study various notions of hyperbolicity for varieties admitting complex polarized variation of Hodge structures (\(\mathbb{C}\)PVHS for short). In the first part we prove that if a quasiprojective manifold \(U\) admits a \(\mathbb{C}\)PVHS whose period map is quasifinite, then \(U\) is algebraically hyperbolic in the sense of Demailly, and that the generalized big Picard theorem holds for \(U\): any holomorphic map from the punctured unit disk to \(U\) extends to a holomorphic map of the unit disk $\Delta$ into any projective compactification of \(U\). This result generalizes a recent work by BakkerBrunebarbeTsimerman. In the second part, we prove the strong hyperbolicity for varieties admitting \(\mathbb{C}\)PVHS, which is analogous to previous works by Nadel, Rousseau, Brunebarbe and Cadorel on arithmetic locally symmetric varieties. In the last part, we show how the techniques developed in this paper yield some new perspectives for hyperbolicity of arithmetic locally symmetric varieties.

Picard theorems for moduli spaces of polarized varieties. (joint work with S. Lu, R. Sun and K. Zuo) arXiv:1911.02973
Abstract
We obtain a general big Picard theorem for the case of complex Finsler pseudometric of negative curvature on logsmooth pairs \((X,D)\). In particular, we show, after a full recall and discussion of the construction of Viehweg and Zuo in their studies of Brody hyperbolicity in the moduli context, that the big Picard theorem holds for the moduli stack \(\mathcal{M}_h\) of polarized complex projective manifolds of semiample canonical bundle and Hilbert polynomial $h$, i.e., for an algebraic variety \(U\), a compactification \(Y\) and a quasifinite morphism \(U \to \mathcal{M}_h\) induced by an algebraic family over \(U\) of such manifolds, that any holomorphic map from the punctured disk \(\mathbb{D}^*\) to \(U\) extends to a holomorphic map \(\mathbb{D} \to Y\). Borel hyperbolicity of \(\mathcal{M}_h\) is then a useful corollary: that holomorphic maps from algebraic varieties to \(U\) are in fact algebraic. We also show the related algebraic hyperbolicity property of \(\mathcal{M}_h\) at the end.

Vanishing theorem for tame
harmonic bundles via \(L^2\)cohomology. (joint work with Feng
HAO) arXiv:1912.02586
Abstract
Using \(L^2\)methods, we prove a vanishing theorem for tame
harmonic bundles over quasiKähler manifolds in a very general
setting. As a special case, we give a completely new proof of the
Kodaira type vanishing theorems for parabolic Higgs bundles due to Arapura et
al. To prove our vanishing theorem, we construct a fine
resolution of the Dolbeault complex for tame harmonic bundles via
the complex of sheaves of \(L^2\)forms, and we establish the
Hörmander \(L^2\)estimate and solve
\((\bar{\partial}_E+\theta)\)equations for the Higgs bundle
\((E,\theta)\).

Hyperbolicity of coarse moduli spaces and isotriviality for certain families. arXiv:1908.08372
Abstract
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized CalabiYau manifolds in the sense of complex Vspaces (a generalization of complex Vmanifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized CalabiYau manifolds, when the Kobayashi pseudodistance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth projective family of polarized CalabiYau manifolds, its variation of the family is less than or equal to the essential dimension of the base.

Hyperbolicity of bases of log CalabiYau
families. arXiv:1901.04423 hal02266744 Abstract
We prove that for any maximally varying, log smooth family of CalabiYau klt pairs, its quasiprojective base is both of log general type, and pseudo Kobayashi hyperbolic. Moreover, such a base is Brody hyperbolic if the family is effectively parametrized.
