Related papers: Unpolarized Shafarevich conjectures for hyper-K\"a…

Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for…

This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to…

We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the…

Artin's conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin's conjecture over fields of characteristic p>3. This implies Tate's conjecture for K3 surfaces over finite…

We prove that, over a smooth quasi-projective curve, the set of non-isotrivial, smooth and projective families of polarized varieties with a fixed Hilbert polynomial and semi-ample canonical bundle is bounded. This extends the boundedness…

We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of K\"ahler metrics satisfying certain nonpositive curvature properties on these compactifications.…

Let $B$ be a smooth projective curve of genus $g$, and $S \subset B$ be a finite subset of cardinality $s$. We give an effective upper bound on the number of deformation types of admissible families of canonically polarized manifolds of…

Proven by A. Parshin and S. Arakelov in the early 70's, Shafaverich hyperbolicity conjecture states that a family of curves of genus $g\ge2$ parametrized by a non hyperbolic curve (\emph{i.e.} isomorphic to $\mathbb{P}^1$, $\mathbb{C}$,…

We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties.…

For a K3 surface over an algebraically closed field of odd characteristic, the representation of the automorphism group on the global two forms is finite. If the K3 surface is supersingular, it is isomorphic to the representation on the…

We show that the classical Kuga-Satake construction gives rise, away from characteristic 2, to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain regular integral model for a Shimura…

We prove an equivariant version of the CM minimization conjecture for extremal K\"ahler manifolds. This involves proving that, given an equivariant punctured family of polarized varieties, a relative version of the CM degree is strictly…

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually…

We prove a dynamical analogue of the Shafarevich conjecture for morphisms $f:\mathbb{P}_K^N\to\mathbb{P}_K^N$ of degree $d\ge2$, defined over a number field $K$. This extends previous work of Silverman and others in the case $N=1$.

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational…

Towards the Lang--Vojta conjecture, we prove results on finiteness and Zariski degeneracy of $S$-integral points of varieties over number fields $k$, including many cases with geometrically irreducible boundary divisors. Our approach builds…

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over finite fields. We prove every K3 surface of finite height over a finite field admits a…

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…

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…

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the…