Organizer: 刘克峰

Speaker: 胡毅（美国亚利桑那大学）

Time: 10:00-11:00, Mar. 17，18    2025

Venue: 至善楼602

Title: Derived and modular resolution of moduli of higher genus stable maps and applications to the reduced GW invariants

Abstract:

This talk is divided into three parts, addressing the following topics:

1. Derived Resolutions and Reduced GW Invariants:

For an integral stack M with a perfect derived object E, there exists a minimal birational modification M′, generally singular, called the derived resolution. Upon pulling E back to M′, E becomes locally diagonalizable, making its 0th sheaf cohomology locally free. Consequently, the Euler class e(E) is well-defined. Applying this construction to the main component of the stable map moduli gives a canonical definition of the reduced Gromov-Witten invariants conjectured by Li-Zinger. This is joint work with Jun Li.

2. Smooth Derived Resolutions in Genus One and Two:

Smooth derived resolutions of stable map moduli exist for genus one and genus two. The genus one case, completed with Jun Li, follows Vakil-Zinger’s construction. The genus two case, a collaboration with Jun Li and Jingchen Niu, relies on explicit local defining equations of the stable map moduli derived by Jun Li and the speaker. Both resolutions can be obtained by blowing up the smooth Artin stack Dg of nodal curve and simple divisor pairs.

3. Stacks with twisted fields and Smooth Derived Resolutions:

Based on the approach in (2), we introduce a framework that birationally modifies a smooth stack M (such as Dg) with tree-like structures (which is modeled on the stable map moduli’s local equations). These modifications ensure that the pullbacks of certain tautological monomial sets possess divisibly minimal elements, enabling smooth derived resolutions. The framework, termed the theory of stacks with twisted fields, provides various smooth derived resolutions for genus two and two canonical resolutions for genus one: one by Vakil-Zinger, followed by Hu-Li, and another obtained by reversing the order of Vakil-Zinger’s virtual blowups, which is novel. This work is in collaboration with Jingchen Niu.