Dori Bejleri
Title: Moduli of surfaces of log general type
Abstract: The Deligne-Mumford-Knudsen space of pointed stable curves is the quintessential example of a compactification of a moduli space. The analogous theory that provides a compactification of the moduli space of log general type surfaces is the theory of KSBA stable pairs (X,D). In the first lecture, I will give the definition of the KSBA moduli space, emphasizing the parallels with and differences from the one dimensional case. The second lecture will focus on the example of fibered surfaces and elliptic surfaces where many examples can be explicitly computed. In the last lecture, I will discuss some general tools and techniques in the theory including wall-crossing, gluing morphisms, and the deformation theory of KSBA stable pairs.
Harold Blum
Title: Moduli of Fano varieties and K-stability
Abstract: K-stability is an algebraic notion introduced by differential geometers to characterize the existence of Kahler-Einstein metrics on Fano varieties. More recently, algebraic geometers have used K-stability to construct a moduli theory for Fano varieties that is often computable in low dimensional examples. In these lectures, I will explain the general theory of K-stability with an emphasis on explicit examples of del Pezzo surfaces and Calabi-Yau pairs.
In the first lecture, I will discuss the definition of K-stability and various useful characterizations. In the second lecture, I will explain the construction of the K-moduli space and focus on explicit examples of del Pezzo surfaces. In the last lecture, I will describe recent work on constructing a moduli theory for CY surface pairs that interpolates between the K and KSBA moduli theories.
Philip Engel
Title: An example-based introduction to compact moduli of K3 surfaces.
Abstract: Due to Torelli theorems, moduli spaces of K3 surfaces are arithmetic quotients of Hermitian symmetric domains. In the 60’s-80’s, compactifications of such varieties were constructed by Baily-Borel, Ash-Mumford-Rapaport-Tai, and Looijenga. But are any of these “semitoroidal” compactifications distinguished, in the sense that they parameterize some "stable" K3 surfaces?
Work on the Minimal Model Program in the 80’s-00’s by Kollár-Shepherd-Barron-Alexeev shows that an ample divisor on a Calabi-Yau variety defines a notion of stability, leading to compact moduli spaces. I will overview joint work with Alexeev, relating the Hodge-theoretic and MMP approaches to compactification.
The first lecture will overview moduli spaces of polarized K3 surfaces, with a running example: degree 2 K3 surfaces. We will discuss the general theory of the period map, Torelli theorem, the Baily-Borel and (semi)toroidal compactifications of Type IV arithmetic quotients. Then we will give explicit descriptions of some of these compactifications, for the degree 2 K3 surfaces.
The second lecture will introduce nice models of degenerations, such as Kulikov and stable models, the notion of a "recognizable divisor", and KSBA compactifications. We will outline the proof that for a recognizable divisor, one gets a semitoroidal compactification of the period space.
Finally, the third lecture will discuss integral-affine structures on the two-sphere, how they encode (in the degree 2 example) the stable K3 surfaces appearing at the boundary of moduli, and how they are used to prove that some explicit guess for the semifan is correct.
Kenneth Ascher
Title:The Hassett-Keel program in genus four
Abstract: The Hassett–Keel program seeks to give a modular interpretation to the steps of the log minimal model program of Mg. I will review the state of the art and discuss recent progress on the Hassett–Keel program in genus four, supplementing earlier results of Casalaina-Martin-Jensen-Laza and Alper–Fedorchuk–Smyth–van der Wyck. Our approach involves wall-crossing for moduli spaces of pairs in the sense of K-stability and KSBA stability, and the recently constructed moduli spaces of boundary polarized Calabi–Yau surface pairs, so I will review these theories. This is based on joint work with Kristin DeVleming, Yuchen Liu, and Xiaowei Wang.
Yuchen Liu
Title: K-moduli of Fano threefolds and genus four curves
Abstract: In this talk, I will show that the K-moduli space of Fano threefolds obtained by blowing up P^3 along (2, 3)-complete intersection curves is isomorphic to a VGIT moduli space studied by Casalaina-Martin-Jensen-Laza. In particular, it is a two-step birational modification of the GIT moduli space of (3, 3)-curves on P^1×P^1. Our strategy is the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients. Based on joint work with Junyan Zhao.
Lisa Marquand
Title: Symplectic birational transformations of Hyperkahler manifolds
Abstract: New examples of compact hyperkahler manifolds are notoriously hard to construct. One approach is to consider a hyperkahler manifold of known type that admits a finite group of symplectic automorphisms, i.e. automorphisms preserving the holomorphic symplectic form. In this case, both the fixed locus and quotient will be symplectic varieties - one hopes to obtain something new (even if singular). The difficulty with this program is twofold: one needs to find a promising group action, and also a geometric example that one can study. In order to do so, often one needs to broaden the search to finite groups of symplectic birational transformations. In this talk I will describe how to first classify possible symplectic group actions for O'Grady's 10-dimensional example, based on joint work with Stevell Muller. I will give examples where it is possible to study the fixed locus for such an action, utilising the construction of Laza, Saccà and Voisin, starting from a cubic fourfold with automorphisms. If time permits, I will report on ongoing work to identify the fixed locus in such an example.