2023 NCTS Higher Dimensional Algebraic Geometry
Mini-courses and Workshop

Title & abstract (video playlist)

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 (video)
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. (video)
Abstract: 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. (video)
Abstract: 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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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 (video)
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.

Title & Abstract (video playlist)
3/20:
Mihnea Popa: Conjectures on the Kodaira Dimension (video)
Alessio Corti: Volume-preserving Birational Maps of P^3 (video)
Hsueh-Yung Lin: Decomposition of the Nef Cones of Fiber Products (video)
Nero Budur: Contact Loci of Arcs (video)
Mihai Paun: Infinitesimal Extension of Pluricanonical Forms and Injectivity (video)

3/21:
James McKernan: The Log Canonical Threshold RevisitedIacopo Brivio (video)
Chenyang Xu: K-stability of Fano Varieties (video)
Kieran O'Grady: Rigid Stable Vector Bundles on Hyperkähler Varieties of type K3^[n] (video)
Hsin-Ku Chen: Minimal Resolutions of Singularities of Threefolds (video)
Alexander Kuznetsov: On 1-nodal Fano Threefolds (video)

3/22:
Paolo Cascini: Foliated Flops over a Complex Threefold (video)
Shinnosuke Okawa: Blowing Down Noncommutative Cubic Surfaces (video)

3/23:
Yongnam Lee: Positivity of the Tangent Bundle of Smooth Projective Surfaces and Fano Threefolds (video)
Iacopo Brivio: Invariance of Plurigenera and KSBA Moduli in Positive and Mixed Characteristic (video)
Joaquin Moraga: Fundamental Groups of low-dimensional Log Canonical Singularities (video)
Tien Cuong Dinh: Positive Closed Currents and Applications in Complex Dynamics (video)
De-Qi Zhang: Kawaguchi–Silverman Conjecture and Zariski Dense Orbit Conjecture (video)

3/24:
Jun-Muk Hwang: Formal Principle with Convergence for Rational Curves of Goursat Type (video)
Burt Totaro: Endomorphisms of Varieties and Bott Vanishing (video)
Kuan-Wen Lai: Nielsen Realization Problem for Bridgeland Stability Conditions on Generic K3 Surfaces (video)
Yuri Prokhorov: On the Classification of Q-Fano Threefolds of Large Fano Index (video)