Elenco seminari del ciclo di seminari
“PROGETTO STRUTTURE: SHAPES, SYMMETRIES AND ARRANGEMENTS”

Moduli of stable sheaves are interesting objects which reflect several properties of underlying spaces. In the theory of algebraic varieties, the canonical bundle is an important invariant. In particular variety with trivial canonical bundle are particulary important. For surfaces, K3 surfaces and abelian surfaces are the examples, and there are many works on these surfaces. Mukai proved that the moduli of stable sheaves on these surfaces are also good by showing they are holomorphic symplectic monifolds. Mukai also invented a quite important tool called Fourier-Mukai transform. With these machinary, he conjectured many interesting phenomena on the moduli spaces in 1980's, and most of them are confirmed affirmatively by Bridgeland's works on Fourier-Mukai transforms and stability conditions. In this lecture, I will explain some results on the moduli of stable sheaves on abelian surfaces including some of Mukai's conjectures: 1. Semi-homogeneous vector bundles and Fourier-Mukai transforms. 2. Stability conditions on abelian surfaces. 3. Moduli of stable sheaves and the Bogomolov factor. 4. Birational invariants and birational automorphisms of moduli spaces.

Orthogonal modular varieties are locally symmetric varieties associated to integral quadratic forms of signature (2,n). They are related to various branches of Mathematics such as Algebraic Geometry (especially as moduli spaces of K3 surfaces and hyperKahler manifolds), Number theory and Representation theory. In these lectures, I will give an introduction to the geometry of orthogonal modular varieties, with focus on the topics such as toroidal compactifications, Kodaira dimension and mixed Hodge structures.

Orthogonal modular varieties are locally symmetric varieties associated to integral quadratic forms of signature (2,n). They are related to various branches of Mathematics such as Algebraic Geometry (especially as moduli spaces of K3 surfaces and hyperKahler manifolds), Number theory and Representation theory. In these lectures, I will give an introduction to the geometry of orthogonal modular varieties, with focus on the topics such as toroidal compactifications, Kodaira dimension and mixed Hodge structures.

Orthogonal modular varieties are locally symmetric varieties associated to integral quadratic forms of signature (2,n). They are related to various branches of Mathematics such as Algebraic Geometry (especially as moduli spaces of K3 surfaces and hyperKahler manifolds), Number theory and Representation theory. In these lectures, I will give an introduction to the geometry of orthogonal modular varieties, with focus on the topics such as toroidal compactifications, Kodaira dimension and mixed Hodge structures.

Part 2: Quadratic line complexes and Kummer surfaces. A quadratic line complex is a nonsingular 3-fold which is the intersection of the Grassmannian G(1,3) (= lines in P^3) and a quadric in P^5. It naturally gives us a Kummer quartic surface S with 16 nodes, a curve C of genus 2, and an abelian surface A. Then A is isomorphic to the Jacobian of C and S is the quotient of A by its inversion. We give a sketch of this classical theory and extend the theory to the case of characteristic 2. Main references are Griffiths, Harris, Principles of Algebraic Geometry, the last chapter and T. Katsura, S. Kondo, arXiv:2301.01450.

Mukai found the relation between polarized symplectic automorphism groups and certain subgroups of the Mathieu group. After the discovery of Mathieu moonshine, Huybrechts established the relation between autoequivalence groups of derived categories of K3 surfaces and certain subgroups of the Conway group. Although we know explicit examples of polarized K3 surfaces with maximal symplectic automorphism groups, it is difficult to find explicit examples of finite autoequivalences of derived categories of K3 surfaces not conjugate to automorphisms of K3 surfaces. In this talk, I would like to study how to construct finite autoequivalences of derived categories of K3 surfaces and discuss their difficulties. First, we recall automorphism groups of compact Riemann surfaces from point of view of algebraic geometry, topology and derived categories. Second, I would like to discuss autoequivalence groups of derived categories of K3 surfaces as an analogue of the case of compact Riemann surfaces.

A Kummer surface was first found by A. Fresnel (1822) for special case and by E. Kummer (1864) for a general one. Later F. Klein (1870) discovered a relation between quadratic line complexes and Kummer surfaces, studied their automorphisms and raised a question on the automorphism group. In this lecture we discuss two topics: Part 1: The Leech lattice and the automorphism group of a generic Kummer surface. By applying the theory of Leech lattice, we present a generator of the automorphism group of a generic Kummer surface associated with a curve of genus 2. This gives an answer of Klein’s question. Main reference is S. Kondo, K3 surfaces, EMS 2020, the last chapter.

Mukai found the relation between polarized symplectic automorphism groups and certain subgroups of the Mathieu group. After the discovery of Mathieu moonshine, Huybrechts established the relation between autoequivalence groups of derived categories of K3 surfaces and certain subgroups of the Conway group. Although we know explicit examples of polarized K3 surfaces with maximal symplectic automorphism groups, it is difficult to find explicit examples of finite autoequivalences of derived categories of K3 surfaces not conjugate to automorphisms of K3 surfaces. In this talk, I would like to study how to construct finite autoequivalences of derived categories of K3 surfaces and discuss their difficulties. First, we recall automorphism groups of compact Riemann surfaces from point of view of algebraic geometry, topology and derived categories. Second, I would like to discuss autoequivalence groups of derived categories of K3 surfaces as an analogue of the case of compact Riemann surfaces.

Mukai found the relation between polarized symplectic automorphism groups and certain subgroups of the Mathieu group. After the discovery of Mathieu moonshine, Huybrechts established the relation between autoequivalence groups of derived categories of K3 surfaces and certain subgroups of the Conway group. Although we know explicit examples of polarized K3 surfaces with maximal symplectic automorphism groups, it is difficult to find explicit examples of finite autoequivalences of derived categories of K3 surfaces not conjugate to automorphisms of K3 surfaces. In this talk, I would like to study how to construct finite autoequivalences of derived categories of K3 surfaces and discuss their difficulties. First, we recall automorphism groups of compact Riemann surfaces from point of view of algebraic geometry, topology and derived categories. Second, I would like to discuss autoequivalence groups of derived categories of K3 surfaces as an analogue of the case of compact Riemann surfaces.