2023 NCTS Higher Dimensional Algebraic Geometry Minicourses and Workshop
March 13 - 24, 2023
Room 515, Cosmology Building, NTU
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3/13-3/17 Mini Course Introduction

Week 1: 3/13-3/17, 2023 Mini Course

3/13-R505, Cosmology Bldg.

3/14-3/17-R515, Cosmology Bldg.

We will have 4 invited lecturers to give mini-courses of 3 lectures each. The total is 12 lectures. The topics of the mini-courses 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 set-up and for foliations.

Lecture II: On foliations of co-rank one on a threefold.

I will give an overview of the Minimal Model Programs for co-rank 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 3-folds

Abstract: I will talk about the classic classification of del Pezzo (smooth Fano) and smooth Fano 3-folds (Iskhoskikh-Prokhorov). 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 Donaldson-Thomas theory, wall-crossing and applications

Lecturer: Yukinobu Toda (Kavli Institute for the Physics and Mathematics of the Universe)

Lecture I: Donaldson-Thomas theory and wall-crossing

Abstract: I will give an overview of Donaldson-Thomas invariants counting points or curves on Calabi-Yau 3-folds and their wall-crossing 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 wall-crossing formula.

Lecture II: Categorical Donaldson-Thomas theory for quivers with super-potentials

Abstract: I will introduce categorical DT theory for quivers with super-potentials, 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 Ext-quivers for DT/PT wall-crossing on Calabi-Yau 3-folds, and give a categorical analogue of DT/PT correspondence via semiorthogonal decomposition. This is a joint work with Tudor Padurariu.

Lecture III: Categorical Donaldson-Thomas 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 quasi-BPS categories, using the result for the DT/PT quiver. This is a joint work with Tudor Padurariu.