Instanton and logarithmic sheaves on threefolds

• Grant number: 22/09063-4
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: October 01, 2022
• End date: September 24, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Marcos Benevenuto Jardim
• Grantee: Gaia Comaschi
• Supervisor: Daniele Faenzi
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Université de Bourgogne, France
• Associated to the scholarship: 19/21140-1 - Moduli spaces of pfaffian representations of cubic three-folds and instanton bundles, BP.PD

Abstract

During my year abroad I plan to devote my research to two main themes: the instanton sheaves on Fano threefolds and the logarithmic tangent sheaves. Instanton sheaves are the subject of the research project that I've conducted in this last year with M. Jardim. In our work we introduced the notion of instanton sheaves on a Fano threefold X of Picard rank one, extending the "classical" one (originally defined for rank 2 vector bundles on the projective space) to non necessarily locally free sheaves of arbitrary rank; we then exhibited several features of these sheaves and investigated their behavior in families. In the specific context of moduli of rank 2 instantons, our results pointed out that a prominent role is played by two objects: the families of non-locally free instantons and the families of curves on X. Through the study of these families, we can indeed get information about the moduli of instanton bundles, allowing, for example, to prove the existence of components, to determine their dimensions and to characterize their boundaries. As it turns out the investigation of families of curves and of non-locally free instantons provides a powerful tool not only to study the moduli of instanton bundles but to determine, more generally, the local and the global aspects of the entire instanton moduli space (since these moduli might contain components parameterizing no locally free sheaves). It is my intent to use the above-mentioned approach to investigate the behavior of families of instantons that still had not been treated by the existing literature. The first is the family of strictly mu-semistable instantons of Fanos of index 2 (these are indeed the only Fano threefolds carrying strictly mu-semistable and Gieseker semistable instantons with non-trivial double dual), and the rank 2 instantons on Fano threefolds of index one (for some of these Fano varieties the existence if instantons is indeed still unknown). The second theme toward which i would like to address my research are the logarithmic tangent sheaves. The definition of logarithmic tangent sheaf, originally introduced on hypersurfaces of the projective space, had recently been extended, by M.Jardim-D.Faenzi and J.Vallees, to complete intersection varieties. One of the main question that we face when we study these sheaves on a complete intersection variety X is determining how the algebraic properties of the regular sequence defining X relate to the sheaf theoretical properties (such as freeness, local freeness or stability) of the associated logarithmic tangent sheaf. An answer to these questions was given, by Faenzi, Jardim, Valles, for the specific case of intersections of 2 quadrics; my goal is to obtain results in these directions for varieties of higher degree of codimensions. Finally I also aim to generalize other results that holds for hypersurfaces, such as the Torelli types theorem, to the complete intersection case. (AU)