3/133/17 Mini Course Introduction
Week 1: 3/133/17, 2023 Mini Course
3/13R505, Cosmology Bldg.
3/143/17R515, Cosmology Bldg.
We will have 4 invited lecturers to give minicourses of 3 lectures each. The total is 12 lectures. The topics of the minicourses focus on derived geometry and birational geometry.
Welcome Reception on Tue evening.
Excursion on Wed afternoon.
Online Recording:https://www.youtube.com/playlist?list=PLQZfZKhc0kiDd8a86V0y3p9nRJsZrdyFN
Invited Lecturers
Paolo Cascini (Imperial College London)
Alexander Kuznetsov (Steklov Mathematical Institute of Russian Academy of Sciences)
Joaquin Moraga (University of California, Los Angeles)
Yukinobu Toda (Kavli Institute for the Physics and Mathematics of the Universe)
Agenda Title & Abstract Download: Please click here to download the file.
A. On the Minimal Model Program for foliations
Lecturer: Paolo Cascini (Imperial College London)
Abstract: The classical Minimal Model Program predicts that a complex projective manifold is either uniruled or it admits a minimal model, i.e. it is birational to a (possibly singular) projective variety whose canonical divisor is nef. The goal of these lectures is to explain an attempt to generalise this Program to the theory of foliations, focusing in particular on three dimensional varieties, on algebraically integrable foliations, and on some applications.
Lecture I: An Introduction to the Minimal Model Program for foliations
Abstract: We will survey some of the main tools in birational geometry, such as the bend and break, the cone theorem and the base point free theorem, both in the classical setup and for foliations.
Lecture II: On foliations of corank one on a threefold.
I will give an overview of the Minimal Model Programs for corank one foliations on a normal complex threefold, with some applications on the study of singularities of such foliations.
Lecture III: Existence of flips.
We will discuss about existence of flips for foliations over a normal complex threefold and for algebraically integrable foliations.
B. Derived categories and rationality of Fano threefolds
Lecturer: Alexander Kuznetsov (Steklov Mathematical Institute of Russian Academy of Sciences)
Abstract: Fano varieties with Picard number 1, or more generally Mori fiber spaces, is the most important class of spaces for the Minimal Model Program of rationally connected varieties. I will survey some classical and recent results in the case of threefolds, concentrating on rationality questions and the structure of derived categories of coherent sheaves.
Lecture I: Smooth Fano threefolds over an algebraically closed field
Abstract: I will overview the classification of Fano threefolds of Picard number 1 over an algebraically closed field with an emphasis on rationality results and the structure of derived categories.
Lecture II: Smooth Fano threefolds over a nonclosed field
Abstract: We will discuss the case of Fano threefolds over nonclosed fields and, more generally, smooth families of Fano threefolds over arbitrary base schemes. In particular, we will discuss rationality criteria for forms of geometrically rational Fano threefolds and the structure of their derived categories.
Lecture III: Conifold transitions
Abstract: We will relate smooth del Pezzo threefolds of degree 1 le d le 5 to smooth prime Fano threefolds of genus g = 2d + 2 passing on the way through maximally nonfactorial nodal prime Fano threefolds. We will discuss some consequences of this construction to derived categories and categorical period maps.
C. Higher dimensional Fano varieties
Lecturer: Joaquin Moraga (University of California, Los Angeles)
Lecture I: Fano surfaces and Fano 3folds
Abstract: I will talk about the classic classification of del Pezzo (smooth Fano) and smooth Fano 3folds (IskhoskikhProkhorov). This will an overview of the known results and a highlight of why understanding Fano varieties is important for Algebraic Geometry.
Lecture II: Kawamata log terminal singularities
Abstract: We will introduce Kawamata log terminal singularities and discuss some classic and new results about this class of singularities. We will explain why understanding these singularities is vital, for instance, through the classification of Gorenstein Fano surfaces of Picard rank one.
Lecture III: Complements on Fano varieties
Abstract: We will discuss the existence of complements on Fano varieties and the boundedness of Fano varieties.
D. Categorical DonaldsonThomas theory, wallcrossing and applications
Lecturer: Yukinobu Toda (Kavli Institute for the Physics and Mathematics of the Universe)
Lecture I: DonaldsonThomas theory and wallcrossing
Abstract: I will give an overview of DonaldsonThomas invariants counting points or curves on CalabiYau 3folds and their wallcrossing formula. I will focus on the MacMahon formula for counting points and DT/PT correspondence for counting curves. I will also give motivations toward categorifications of DT invariants and wallcrossing formula.
Lecture II: Categorical DonaldsonThomas theory for quivers with superpotentials
Abstract: I will introduce categorical DT theory for quivers with superpotentials, and explain basic tools of its study, e.g. window theorem, categorical Hall products. I will then focus on specific quivers called DT/PT quivers, which appear as Extquivers for DT/PT wallcrossing on CalabiYau 3folds, and give a categorical analogue of DT/PT correspondence via semiorthogonal decomposition. This is a joint work with Tudor Padurariu.
Lecture III: Categorical DonaldsonThomas theory for local surfaces
Abstract: I will give a definition of categorical DT theory for local surfaces, i.e. the total spaces of canonical line bundles on surfaces, based on Koszul duality and singular support theory. I will then give semiorthogonal decompositions of DT categories into PT categories and quasiBPS categories, using the result for the DT/PT quiver. This is a joint work with Tudor Padurariu.
