Searcharxiv⌕ Search

arXiv subjects

Dominique Mattei

Publications and source records attributed to Dominique Mattei.

10 recordsLinked to original sources
arXiv2026

Finite order symplectic birational self-maps on Kummer-type manifolds

A projective hyperk\"ahler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of Picard rank 3, any projective Kummer-type manifold admitting a finite-order symplectic birational self-map that acts nontrivially on its second cohomology group is twisted modular. We provide a complete characterization of these exceptions in terms of their N\'eron-Severi lattices. We then investigate symplectic birational self-maps of modular Kummer-type manifolds, determining exactly which Mukai vectors allow the birational transformation induced by crossing the vertical wall, which acts on cohomology as a reflection, to correspond to a finite-order symplectic birational self-map. Additionally, we prove in an appendix several results concerning moduli spaces of twisted sheaves on abelian surfaces which were not readily available in the literature.

math.AG↗
arXiv2025

The Tate-Shafarevich group of a polarised K3 surface

In an earlier paper we generalised the notion of the Tate-Shafarevich group of an elliptic K3 surface to the Tate-Shafarevich group of a polarised K3 surface. In the present note, we complement the result by proving that the Tate-Shafarevich group of a polarised K3 surface (S,h) with h primitive parametrises bijectively all torsors for the Jacobian of the generic curve in the linear system |h| that admit a good hyperk\"ahler compactification. The result is seen as the analogue of the classical fact that the Tate-Shafarevich group of an elliptic K3 surface is the subgroup of the Weil-Ch\^atelet group of all twists that can be compactified to a K3 surface.

math.AG↗
arXiv2024

Twists of intermediate Jacobian fibrations

We study the sections, Tate--Shafarevich twists, and the period for an OG10 hyperk\"ahler Lagrangian associated to a cubic fourfold. To do so, we introduce the analytic relative Jacobian sheaf for a Lagrangian fibration of a hyperk\"ahler variety. The Tate--Shafarevich group parameterizing twists is isomorphic to the first cohomology group of this sheaf and we compute it in terms of certain analytic Brauer groups associated to the cubic fourfold. We prove that the primitive Hodge lattice of the cubic fourfold is, up to a sign, isometric to a distinguished sublattice of the second cohomology group of the associated OG10 hyperk\"ahler manifold. Among the main tools we use are intersection complexes with integral coefficients, Decomposition Theorem, Hodge modules and Deligne cohomology.

math.AG↗
arXiv2024

Obstruction classes for moduli spaces of sheaves and Lagrangian fibrations

We investigate obstruction classes of moduli spaces of sheaves on K3 surfaces. We extend previous results by Caldararu, explicitly determining the obstruction class and its order in the Brauer group. Our main theorem establishes a short exact sequence relating the Brauer group of the moduli space to that of the underlying K3 surface. This provides a criterion for when the moduli space is fine, generalising well-known results for K3 surfaces. Additionally, we explore applications to Ogg-Shafarevich theory for Beauville-Mukai systems. Furthermore, we investigate birational equivalences of Beauville-Mukai systems on elliptic K3 surfaces, presenting a complete characterisation of such equivalences.

math.AG↗
arXiv2023

The special Brauer group and twisted Picard varieties

We generalise the notion of the Tate-Shafarevich group of an elliptic K3 surface with a section to the Tate-Shafarevich group of a K3 surface endowed with a linear system. The construction, which uses Grothendieck's special Brauer group, provides an efficient way to deal with moduli spaces of twisted sheaves supported on curves in a K3 surface.

math.AG↗
arXiv2023

The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties

We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold $X$ containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to $X$ up to a cover. To do so, we propose a general construction of Lagrangian fibrations in Prym varieties as subfibrations of Beauville-Mukai systems over some loci of nodal curves in linear systems on K3 surfaces.

math.AG↗
arXiv2022

On symplectic birational self-maps of projective hyperk\"{a}hler manifolds of K3$^{[n]}$-type

We prove that projective hyperk\"{a}hler manifolds of K3$^{[n]}$-type admitting a non-trivial symplectic birational self-map of finite order are isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3 surfaces. Motivated by this result, we analyze the reflections on the movable cone of moduli spaces of sheaves and determine when they come from a birational involution.

math.AG↗
arXiv2020

Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9

A complex smooth prime Fano threefold $X$ of genus $9$ is related via projective duality to a quartic plane curve $\Gamma$. We use this setup to study the restriction of rank $2$ stable sheaves with prescribed Chern classes on $X$ to an anticanonical $K3$ surface $S\subset X$. Varying the threefold $X$ containing $S$ gives a rational Lagrangian fibration $$\mathcal{M}_S(2,1,7) \dashrightarrow \mathbb{P}^3$$ with generic fibre birational to the moduli space $\mathcal{M}_X(2,1,7)$ of sheaves on $X$. Moreover, we prove that this rational fibration extends to an actual fibration on a birational model $\mathcal{M}$ of $\mathcal{M}_S(2,1,7)$. In a last part, we use Bridgeland stability conditions to exhibit all $K$-trivial smooth birational models of $\mathcal{M}_S(2,1,7)$, which consist in itself and $\mathcal{M}$. We prove that these models are related by a flop, and we describe the positive, movable and nef cones of $\mathcal{M}_S(2,1,7)$.

math.AG↗
arXiv2019

Categorical vs topological entropy of autoequivalences of surfaces

In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.

math.AG↗