My research subjects include complex algebraic-analytic geometry, complex hyperbolicity, non-abelian Hodge theories (both Archimedean and non-Archimedean settings), Nevanlinna theory, and their interactions.
My current interests focus on harmonic mappings to Euclidean buildings, the linear Shafarevich conjecture, the Chern-Hopf-Thurston conjecture (sign of Euler characteristic), Kollár's conjecture on positivity of holomorphic Euler characteristic, hyperbolicity of algebraic varieties via representations of fundamental groups,
deformation invariance of \(\Gamma\)-dimension (conjecture by Claudon and Campana), and algebro-geometric properties of varieties with large/big fundamental groups.
Using \(L^2\)-methods, we prove a vanishing theorem for tame
harmonic bundles over quasi-Kähler 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
\((E,\theta)\).
In this paper we study various hyperbolicity properties for a quasi-compact 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 quasi-projective 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.
In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson.
Several byproducts are also obtained in this paper.
For maximal variational smooth families of projective manifolds whose general fibers have semi-ample 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 Campana-Paun and later generalized by Popa-Schnell. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective 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 semi-ample canonical bundle. This proves a conjecture by Viehweg-Zuo 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 To-Yeung, in which they further assumed that these families are canonically polarized.
We obtain a general big Picard theorem for the case of complex Finsler pseudometric of negative curvature on log-smooth 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 semi-ample canonical bundle and Hilbert polynomial $h$, i.e., for an algebraic variety \(U\), a compactification \(Y\) and a quasi-finite 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.
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.
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 Diverio-Trapani on the ampleness of jet bundles of general complete intersections in complex projective spaces.
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 n-dimensional 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
Brotbek-Darondeau link and Xie independently.
In the first part of the paper, we study a Fujita-type 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 Demailly-Peternell-Schneider 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.
The main goal of this article is to construct "generalized Okounkov
bodies" for an arbitrary transcendental pseudo-effective (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.
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.
In 2001, de Oliveira, Katzarkov, and Ramachandran conjectured that the property of smooth projective varieties having big fundamental groups is stable under small deformations. This conjecture was proven by Benoît Claudon in 2010 for surfaces and for threefolds under suitable assumptions. In this paper, we prove this conjecture for smooth projective varieties admitting a big complex local system. Moreover, we address a more general conjecture by Campana and Claudon concerning the deformation invariance of the \(\Gamma\)-dimension of projective varieties. As an application, we establish the deformation openness of pseudo-Brody hyperbolicity for projective varieties endowed with a big and semisimple complex local system. To achieve these results, we develop the deformation regularity of equivariant pluriharmonic maps into Euclidean buildings and Riemannian symmetric spaces in families, along with techniques from the reductive and linear Shafarevich conjectures.
Given a complex smooth quasi-projective variety \(X\), a reductive algebraic group \(G\) defined over some non-archimedean local field \(K\) and a Zariski dense representation \(\varrho:\pi_1(X)\to G(K)\), we construct a \(\varrho\)-equivariant pluriharmonic map from the universal cover of \(X\) into the Bruhat-Tits building \(\Delta(G)\) of \(G\), with appropriate asymptotic behavior. We also establish the uniqueness of such a pluriharmonic map in a suitable sense, and provide a geometric characterization of these equivariant maps.
This paper builds upon and extends previous work by the authors jointly with G. Daskalopoulos and D. Brotbek.
In 1995, Kollár conjectured that a complex projective \(n\)-fold \(X\) with generically large fundamental group has Euler characteristic \(\chi(X, K_X)\geq 0\). In this paper, we confirm the conjecture assuming \(X\) has linear fundamental group, i.e., there exists an almost faithful representation \(\pi_1(X)\to {\rm GL}_N(\mathbb{C})\). We deduce the conjecture by proving a stronger \(L^2\) vanishing theorem: for the universal cover \(\widetilde{X}\) of such \(X\), its \(L^2\)-Dolbeaut cohomology \(H_{(2)}^{n,q}(\widetilde{X})=0\) for \(q\neq 0\). The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.
If \(X\) is a closed \(2n\)-dimensional aspherical manifold, i.e., the universal cover of \(X\) is contractible, then the Chern-Hopf-Thurston conjecture predicts that \((-1)^n\chi(X)\geq 0\). We prove this conjecture when \(X\) is a complex projective manifold whose fundamental group admits an almost faithful linear representation over any field. In fact, we prove a much stronger statement that if \(X\)is a complex projective manifold with large fundamental group and \(\pi_1(X)\) admits an almost faithful linear representation, then \(\chi(X, \mathcal{P})\geq 0\) for any perverse sheaf \(\mathcal{P}\) on \(X\).
To prove the main result, we introduce a vanishing cycle functor of multivalued one-forms. Then using techniques from non-abelian Hodge theories in both archimedean and non-archimedean settings, we deduce the desired positivity from the geometry of pure and mixed period maps.
Given a complex quasi-projective normal variety \(X\) and a linear representation \(\rho:\pi_1(X)\to {\rm GL}_N(K)\) with \(K\) any field of positive characteristic, we mainly establish the following results:
1. the construction of the Shafarevich morphism associated with \(\rho\).
2. In cases where \(X\) is projective, \(\rho\) is faithful and the \(\Gamma\)-dimension of X is at most two (e.g. \({\rm dim} X=2\)), we prove that the Shafarevich conjecture holds for X.
3. In cases where \(\rho\) is big, we prove that the Green-Griffiths-Lang conjecture holds for X.
4. When \(\rho\) is big and the Zariski closure of \(\rho(\pi_1(X))\) is a semisimple algebraic group, we prove that \(X\) is pseudo Picard hyperbolic, and strongly of log general type.
5. If \(X\) is special or \(h\)-special, then \(\rho(\pi_1(X))\) is virtually abelian.
We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.
In this paper, we present a more accessible proof of Eyssidieux's proof of the reductive Shafarevich conjecture in 2004, along with several generalizations. In a nutshell, we prove the holomorphic convexity of the covering of a projective normal variety \(X\), which corresponds to the intersection of kernels of reductive representations \(\varrho:\pi_1(X)\to {\rm GL}_N(\mathbb{C})\). Our approach avoids the necessity of using the reduction mod p method employed in Eyssidieux's original proof. Moreover, we extend the theorems to singular normal varieties under a weaker condition of absolutely constructible subsets, thereby answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran. Additionally, we construct the Shafarevich morphism for reductive representations over quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level.
It is natural to ask how fundamental groups \(\pi_1(X)\) of complex quasi-projective varieties \(X\) determine their hyperbolicity. In this paper we address this question when there is a linear representation \(\varrho:\pi_1(X)\to GL_N(\mathbb{C})\). First, if $\varrho$ is big and the Zariski closure of \(\varrho\big(\pi_1(X)\big)\) in \(GL_N(\mathbb{C})\) is a semisimple algebraic group, then we prove that there is a proper Zariski closed subset \(Z\subsetneqq X\) such that any closed irreducible subvariety \(V\) of \(X\) not contained in $Z$ is of log general type; any holomorphic map from the punctured disk \(\Delta^*\) to \(X\) with image not contained in \(Z\) does not have essential singularity at the origin. In particular, \(X\) is of log general type and all entire curves in $X$ lie on \(Z\). We provide examples to illustrate that the condition is sharp.
Second, if \(X\) is a special or hyperbolically special quasi-projective manifold in the sense of Campana, we prove that \(\varrho\big(\pi_1(X)\big)\) is virtually nilpotent. We also construct examples to show that this result is optimal, thus disproving Campana's abelianity conjecture for quasi-projective manifolds.
To prove the above theorems we develop new feature in non-abelian Hodge, geometric group and Nevanlinna theories. Along the way we prove the strong Green-Griffiths-Lang conjecture for quasi-projective varieties \(X\) admitting a morphism \(a:X\to A\) with \(\dim X=\dim a(X)\)
where \(A\) is a semi-abelian variety; a reduction theorem for Zariski dense representations \(\pi_1(X)\to G(K)\) where \(G\) is a reductive group over a non-Archimedean local field \(K\).
In this paper we develop some non-abelian Hodge techniques over complex quasi-projective manifolds \(X\) in both archimedean and non-archimedean contexts. In the non-archimedean case, we first generalize a theorem by Gromov-Schoen: for any Zariski dense representation \(\rho:\pi_1(X)\to G(K)\), where \(G\) is a semisimple algebraic group defined over some non-archimedean local field $K$, we construct a \(\rho\)-equivariant harmonic map from \(X\) into the Bruhat-Tits building \(\Delta(G)\) of \(G\) with some suitable asymptotic behavior. We then construct logarithmic symmetric differential forms over \(X\) when the image of such \(\rho\) is unbounded. Our main result in the archimedean case is that any semisimple representation \(\sigma: \pi_1(X)\to\operatorname{GL}_N(\mathbb{C})\) is rigid provided that \(X\) does not admit logarithmic symmetric differential forms. Furthermore, such representation \(\sigma\) is conjugate to \(\sigma':\pi_1(X)\to\operatorname{GL}_N(\mathcal{O}_L)\) where \(\mathcal{O}_L\) is the ring of integer of some number field \(L\), so that \(\sigma'\) is a complex direct factor of a \(\mathbb{Z}\)-variation of Hodge structures.
As an application we prove that a complex quasi-projective manifold \(X\) has nonzero global logarithmic symmetric differential forms if there is linear representation \(\pi_1(X)\to\operatorname{GL}_N(\mathbb{C})\) with infinite images.
For a quasi-compact Kähler manifold \(U\) endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that \(U\) is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover \(\tilde{U}\) of \(U\) from a quasi-projective manifold \(\tilde{U}\) so that any projective compactification of \(\tilde{U}\) is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbolicity for quasi-compact Kähler manifolds.
Let \(f: Y\to X\) be a morphism between smooth complex quasi-projective varieties and \(Z\) be the closure of \(f(Y)\) with \(\iota: Z\to X\) the inclusion map. We prove that:
1. for any field \(K\), there exist finitely many semisimple representations \(\{\tau_i:\pi_1(Z)\to {\rm GL}_N(\overline{k})\}_{i=1,\ldots,\ell}\) with \(k\subset K\) the minimal field contained in \(K\) such that if \(\varrho:\pi_1(X)\to {\rm GL}_{N}(K)\) is any representation satisfying \([f^*\varrho]=1\), then \([\iota^*\varrho]=[\tau_i]\) for some \(i\).
2. The induced morphism between \({\rm GL}_N\)-character varieties (of any characteristic) of \(\pi_1(X)\) and \(\pi_1(Y)\) is quasi-finite if \({\rm Im}[\pi_1(Z)\to \pi_1(X)]\) is a finite index subgroup of \(\pi_1(X)\).
These results extend the main results by Lasell in 1995 and Lasell-Ramachandran in 1996 from smooth complex projective varieties to quasi-projective cases with richer structures.
Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold \(V\). When the associated moduli map \(V\to P_h\) from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold \(V\): for any projective compactification \(Y\) of \(V\), any holomorphic map \(f:\Delta-\{0\}\to V\) from the punctured unit disk to \(V\) extends to a holomorphic map of the unit disk \(\Delta\) into \(Y\). This result generalizes our previous work on the Brody hyperbolicity of \(V\) (i.e. there are no entire curves on \(V\)), as well as a more recent work by Lu-Sun-Zuo on the Borel hyperbolicity of \(V\) (i.e. any holomorphic map from a quasi-projective variety to \(V\) is algebraic). We also obtain generalized big Picard theorem for bases of log Calabi-Yau families.
In this paper we prove that every quasi-projective base space \(V\) of
smooth family of minimal projective manifolds with maximal variation is
pseudo Kobayashi hyperbolic, that is, \(V\) is Kobayashi hyperbolic modulo a
proper subvariety \(Z\subsetneq V\). In particular, every nonconstant
entire curve \(f:\mathbb{C}\to V\) has image \(f(\mathbb{C})\) on
a proper subvariety \(Z\subsetneq V\). As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which prove a conjecture by Viehweg-Zuo in 2003.
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex V-spaces (a generalization of complex V-manifolds 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 Calabi-Yau manifolds, when the Kobayashi pseudo-distance 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 Calabi-Yau manifolds, its variation of the family is less than or equal to the essential dimension of the base.
We prove that for any maximally varying, log smooth family of Calabi-Yau klt pairs, its quasi-projective base is both of log general type, and pseudo Kobayashi hyperbolic. Moreover, such a base is Brody hyperbolic if the family is effectively parametrized.
Séminaire de géométrie complexe
Date: 2 Oct 2025
Location: Institut de Mathématiques de Toulouse, Toulouse
Title: Topology of complex algebraic varieties
COMPLEX GEOMETRY AND BEYOND
Date: May 31 -June 4 2025
Location: East China Normal University, Shanghai, China
Title: Deformation of Algebraic Varieties with Big Fundamental Groups
Séminaire de géométrie complexe
Date: 24 Mar 2025
Location: IECL, Nancy
Title: Deformation of Varieties with Big Fundamental Groupss
Workshop on Complex Geometry
Date: 24-27 October, 2023
Location: Hong Kong
Title: Constructing Shafarevich morphism for representations in positive characteristic and hyperbolicity
