Related papers: Linear Shafarevich Conjecture in positive characte…

This paper investigates the relationship between the hyperbolicity of complex quasi-projective varieties $X$ and the (topological) fundamental group $\pi_1(X)$ in the presence of a linear representation $\varrho: \pi_1(X) \to {\rm…

This is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: \pi_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois…

This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we…

This paper is Part III of a series of three. We begin by introducing the notion of $h$-special varieties, which can be seen as varieties "chain-connected by the Zariski closures of entire curves." We prove that if $X$ is either a special…

In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $\rho:\pi_1(X)\to {\rm GL}_{N}(\mathbb{C})$,…

The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using…

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…

This is the first part of a series of three papers. In this paper, we establish a Big Picard type theorem for holomorphic maps $f:Y \to X$, where $Y$ is a ramified covering of the punctured disc $\mathbb{D}^*$ with small ramification and…

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…

The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over C, there exists a proper algebraic subvariety of X containing all non constant entire curves f : C $\rightarrow$ X. Using the formalism…

We extend to compact K\"ahler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach based on an interversion lemma for fibrations with tori versus general type…

The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -- especially through the concepts of curvature and positivity which are central…

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 classical Shafarevich conjecture predicts that the universal cover of a complex smooth projective variety $X$ is holomorphically convex. In this paper, we propose a refinement of this conjecture for varieties defined over the reals. In…

This article is devoted to examples of (orbifold) K\"ahler groups from the perspective of the so-called Shafarevich conjecture on holomorphic convexity. It aims at pointing out that every quasi-projective complex manifold with an…

We show the following algebraicity result for a complex projective variety $X$ with big representation of $\pi_1$ into a semi-simple algebraic group: There exists a proper subvariety $Z \subset X$ such that for any algebraic curve $C$, any…

Let f: X -> Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has…

Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture…

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 a. for any field $K$, there exist finitely many semisimple…

In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. First, we study theories of proper hyperbolic polycurves over regular schemes.…