Related papers: Topology, Hyperbolicity, and the Shafarevich Conje…

We describe in some details an idea of M. Kontsevich how one can try to find a counterexample to the Hodge conjecture using tropical geometry.

A Tate-Shafarevich twist $X^\phi\to B$ of a fibration $X\to B$ modifies it by a $1$-cocycle of flows of vector fields relative to the base, locally in the analytic topology. Sacc\`a conjectured that the total spaces of two projective…

We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire…

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of…

Motivated by conjectures of Demailly, Green-Griffiths, Lang, and Vojta, we show that several notions related to hyperbolicity behave similarly in families. We apply our results to show the persistence of arithmetic hyperbolicity along field…

We obtain a lower estimate for the Hilbert series of Jacobi algebras and their completions by providing analogue of the Golog-Shafarevich-Vinberg theorem for potential case. We especially treat non-homogeneous situation. This estimate…

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by…

The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold has an almost…

We prove an analogue for algebraic stacks of Hermite-Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley-Weil type theorem for integral points on stacks. As an application of our results, we prove…

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

We introduce a new class of gauge field theories in any complex dimension, based on algebra-valued (p,q)-forms on complex n-manifolds. These theories are holomorphic analogs of the well-known Chern-Simons and BF topological theories defined…

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable…

We consider algebraic varieties canonically associated to any Lie superalgebra, and study them in detail for super-Poincar\'e algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of)…

We study the algebraic hyperbolicity of the complement of very general degree $2n$ hypersurfaces in P^n. We prove the Algebraic Green-Griffiths-Lang Conjecture for these complements, and in the case of the complement of a quartic plane…

We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that…

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology…

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is…

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the…

We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double…

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega_{X} \otimes…