We will begin by reviewing the classical moduli theory of curves developed by Deligne–Mumford and Grothendieck–Knudsen. We will then discuss its higher dimensional analogues, i.e., the KSBA moduli theory. A central issue is that flatness behaves differently in higher dimensions: it is too weak for the underlying variety, but too strong for the divisorial part. Finally, using an arrangement of lines as an example, we will illustrate how the Minimal Model Program (MMP) can be used to find stable degenerations.

Several KSBA compactifications have been constructed explicitly, with their families (all stable pairs) described in detail. Examples include pairs of plane curves with small coefficients, principally polarized abelian varieties (PPAVs), hyperplane arrangements, K3 surfaces with recognizable divisors and so on. We will briefly review some of these constructions then focusing on hyperplane arrangements, where we will discuss the two different approaches developed by Hacking–Keel–Tevelev via visible contours and by Alexeev via geometric invariant theory (GIT). Finally, we will propose several problems for graduate students to explore. 

In 1966, Hartshorne proved that Hilbert schemes of projective space with a fixed Hilbert polynomial are connected. In contrast, Seong (2020) established disconnectedness of certain Hilbert schemes of ordinary Grassmannians and computed their nef cones in the hypersurface setting. Motivated by these results, this talk explores analogous questions for orthogonal Grassmannians. We will begin with a crash course in Schubert calculus, introducing Schubert varieties and the classical Pieri rule. We will then discuss Hilbert schemes parametrizing hypersurfaces in linear spaces contained in orthogonal Grassmannians, explaining how to describe their connected components and determine their nef cones.

Title : As a continuation of the previous talk, we will discuss the construction of the KSBA moduli spaces and the universal families for hyperplane arrangements. Kapranov’s version of Gelfand—MacPherson correspondence provides a natural candidate for the moduli space. Subsequently, Hacking—Keel—Tevelev and Alexeev constructed the corresponding families using matroid polytope tilings, via different approaches (visible contour and GIT respectively). We will also discuss wall-crossing problems in KSBA moduli theory, with particular emphasis on the case of hyperplane arrangements.

Holomorphic symplectic varieties are among the building blocks of varieties with trivial canonical class, and are among the central objects in geometric representation theory. Many of them arise as moduli spaces of objects on two-dimensional Calabi–Yau spaces (in a broad sense), such as representations of the preprojective algebra of a finite quiver, semistable sheaves and principal bundles on a K3 surface and more. At the points parametrizing ‘non-simple’ objects these moduli spaces are singular in general, and their local structures have been investigated through careful investigation of each moduli problem by  Arbarello–Saccà, Budur–Zhang, Davison and more. In all cases, the result is that the moduli spaces are complex-analytically-locally isomorphic to explicit linear hyperkähler quotients, which hints the existence of a uniform proof applicable to any ‘moduli on two-dimensional Calabi-Yau spaces’. In this talk we will explain such a proof, whose key idea is to utilize an extension of symplectic structure to a space that also remembers the symmetries of the objects, in such a way that the singularities and the symmetries are paired. This is the so-called 0-shifted symplectic structure, which we will explain through examples. Based on joint work in progress with Young-Hoon Kiem and Hyeonjun Park.

In this lecture, I will introduce Gromov–Witten invariants and explain from the basics how they can be computed using localization techniques. As an application, I will also briefly discuss the proofs of the holomorphic anomaly equations for Gromov–Witten invariants of local P2 and the quintic threefold.

- Kyoung-Seog Lee (10:30)
- Rodolfo Aguilar Aguilar (2:00)