数学研究所

学术报告

代数几何研讨班

Speaker: 潘龙（复旦大学）

Inviter: 刘杰 副研究员

Language: Chinese

Title: K-moduli space of log del pezzo surfaces

Time&Venue: 2025年3月11日（周二） 10：00-11：30 & 晨兴410

Abstract: The concept of K-stability arises from the existence of Kähler-Einstein metrics onFano manifolds in complex geometry. Recently, algebraic geometers have used tools from K-stability theory to construct compact moduli spaces of Fano varieties; however, the study of specific examples remains poorly understood.Even for certain surface pairs, the wall-crossing phenomenon remains unclear.In this talk, I will first give a brief review of these theories. We will then apply the wall-crossing theory of K-moduli spaces to study the K-moduli spacesof two classes of log del Pezzo surface pairs.

The main results consist of two parts. The first isthe study of the K-moduli space of degree 8 del Pezzo surface pairs (F_1, C)where C ∈|−2K_{F_1}|.This moduli space is closely related to the moduli space of certain K3 surfaces with anti-symplectic involutions. We use tools involving normalized volumes and T-singularity theory to determine all possible degenerations on the boundary.We then employ the theory of complexity-one torus actions to calculate the positions of critical walls and classify critical surface pairs. Finally, we describe the wall-crossing behavior of this K-moduli space via the local VGIT presentation. Notably,our moduli space provides the first example of a K-moduli space that can be associated with non-reductive GIT.

The second part concerns the study of the moduli spaceof degree 9 del Pezzo surfaces with multiple boundaries (P^2; aQ + bL), where Q is a plane quintic and L is a line. This provides the first example of higher-dimensional wall-crossing in K-moduli spaces. This K-moduli space has anatural connection with the VGIT moduli space of degree 5 pairs investigated by Laza. We prove that our K-moduli space is isomorphic to Laza’s moduli space when the coefficients of the surface pairs satisfy specific constraints.