Absolute intersection motive
The purpose of this article is to define and study the notion of absolute intersection motive.
Motifs : un tour d'horizon
Ce texte constitue une introduction (partielle et partiale) à la théorie des motifs, un programme initié par Grothendieck et qui structure aujourd'hui la géométrie et l'arithmétique des variétés algébriques.
LINEAR SHAFAREVICH CONJECTURE IN POSITIVE CHARACTERISTIC, HYPERBOLICITY AND APPLICATIONS
Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho:\pi_1(X)\to {\rm GL}_{N}(K)$ with $K$ any field of positive characteristic, we mainly establish the following results: \begin{enumerate}[label*={\rm (\alph*)}] \item the construction of the Shafarevich morphism ${\rm sh}_\varrho:X\to {\rm Sh}_\varrho(X)$ associ...
VANISHING THEOREM AND A CONJECTURE OF KOLLÁR
In 1995, Koll\'ar 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 deduc...
LINEAR CHERN-HOPF-THURSTON CONJECTURE
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. I...
Anti-symplectic involutions on moduli spaces of sheaves on K3 surfaces via auto-equivalences
We provide new examples of anti-symplectic involutions on moduli spaces of stable sheaves on K3 surfaces. These involutions are constructed through (anti) autoequivalences of the bounded derived category of coherent sheaves on K3 surfaces arising from spherical bundles. We analyze these induced maps in the moduli space, imposing restrictions on the...
RIGIDITY OF PERIODIC POINTS FOR LOXODROMIC AUTOMORPHISMS OF AFFINE SURFACES
We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same periodic points. We construct canonical heights for these automorphisms and use arithmetic equidistribution for adelic line bundles over quasiprojective varieties following t...
On {\L}ojasiewicz Inequalities and the Effective Putinar's Positivstellensatz
The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the de...
Intersection of orbits of loxodromic automorphisms of affine surfaces
We show the following result: If X_0 is an affine surface over a field K and f , g are two loxodromic automorphisms with an orbit meeting infinitely many times, then f and g must share a common iterate. The proof uses the preliminary work of the author in [Abb23] on the dynamics of endomorphisms of affine surfaces and arguments from arithmetic dyna...
Isolated singularities, inverse systems and the punctual Hilbert scheme
We study of the punctual Hilbert scheme from an algorithmic point of view. We first present algorithms, which allow to compute the inverse system of an isolated point. We define the punctual Hilbert scheme as a subvariety of a Grassmannian variety and provide explicit equations defining it. Then we localised our study to the algebraic variety Hilb_...
