In this thesis we study sheaves on hyper-Kähler manifolds, with final goal to solve a long standing conjecture due to Markman and O'Grady. Namely, we show that we can realize OG10 as a moduli space of stable bundles on a hyper-Kähler fourfold. We begin with stable sheaves and, more in general, stable complexes on K3 surfaces. It is a celebrated res...
An affine surface (over an algebraically closed field) is a variety of dimension 2 defined by polynomial equations. Given an endomorphism of such a surface, we can ask the following questions: Are there Zariski dense orbits ? If the orbit of a point goes to infinity, can we control the speed of divergence ? Are there a lot of periodic orbits ? Can ...
We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus ggeq4. For a given Riemann matrix \Omega\in\mathbb{H}^g , the Fay identity establishes linear dependence of secants in the Kummer variety if and only if the Riemann matrix corresponds to a Jacobian variety as shown by Krichever. The the...
This thesis is situated at the interface between theoretical physics, in particular string theory and supersymmetric field theories, and mathematics, in particular algebraic geometry and representation theory. We study the counting of BPS ie supersymmetry-preserving) states in four-dimensional N=2 supersymmetric theories, obtained by compactifying ...
This thesis deals with the coherent cohomology and geometry of certain Shimura varieties in prime characteristic p. More precisely, we consider Siegel varieties with a hyperspecial level at p. We start by establishing positivity results for some automorphic bundles. These results require the introduction of a new positivity notion for vector bundle...
We try to find a basis of the numerical Grothendieck group G of very general Küchle fourfolds of type c5, denoted X₄. To do so we first study the geometry of such varieties, and try to deduce from this a family F of elements in G which could be a basis. Then we try to compute the matrix whose coefficients are Euler characteristics between elements ...
L’objectif de cette thèse est d’approfondir les liens mis en lumière par McDuff-Polterovic, Biran et Opshtein entre l’existence de plongements de domaines symplectiques dans une variété symplectique et l’existence de courbes symplectiques réalisant des singularités algébriques planes isolées dans ces variétés. Pour ce faire, on rappelle dans un pre...
Nous énonçons des résultats de descente relative à la topologie sur les espaces de Berkovich dont les flèches couvrantes sont plates et surjectives. Nous donnons des conditions suffisantes pour qu’une catégorie fibrée donnée soit un champs pour cette topologie. Ensuite, nous utilisons ce résultat pour montrer que le foncteur tiré en arrière de la c...
Probabilistic proof systems, such as probabilistically checkable proofs, interactive proofs, and zero-knowledge proofs, feature the common characteristic of having a probabilistic verification procedure. Notably, such proof systems are at the heart of cryptographic protocols that enable polylogarithmic-time verification of very long computations. G...
In this thesis we study some complete linear systems associated to divisors of Hilbert schemes of 2 points on complex projective K3 surfaces with Picard group of rank 1, together with the rational maps induced. We call these varieties Hilbert squares of generic K3 surfaces, and they are examples of irreducible holomorphic symplectic (IHS) manifold....
