The virtual fundamental class for the moduli space of surfaces of general type

TL;DR

The paper constructs a Behrend–Fantechi–Li–Tian compatible virtual fundamental class for KSBA moduli spaces of semi-log-canonical surfaces of general type by introducing and exploiting the moduli stack of locally complete intersection (lci) covers and index-one covers. It proves that the lci-cover moduli stack $M^{\mathrm{lci}}$ carries a perfect obstruction theory and that the induced virtual class descends to the KSBA moduli stack $M$, thereby validating Donaldson’s conjecture for the existence of a virtual fundamental class in this setting. A tautological projective/invariant theory is developed via the CM line bundle, defining invariants $I_{\mathrm{CM}}$ by integrating powers of $c_1(L_{\mathrm{CM}})$ against the virtual class, thereby generalizing Miller–Morita–Mumford-type invariants from stable curves to stable surfaces. The work also analyzes smoothing components and equivariant deformations, providing concrete examples (e.g., quintic and sextic surfaces) to illustrate the construction and to connect to broader themes such as mirror symmetry and log Calabi–Yau moduli. Overall, the framework offers a robust toolkit to study enumerative invariants on moduli of surfaces of general type and bridges KSBA compactifications with virtual geometry techniques.

Abstract

We propose a construction of an obstruction theory on the moduli stack of index-one covers of semi-log-canonical surfaces of general type. Using the index-one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ cover. The $\lci$ cover, as a Deligne-Mumford stack, has only locally complete intersection singularities. We then construct the moduli stack of $\lci$ covers so that it admits a proper map to the moduli stack of surfaces of general type. Next, we construct a perfect obstruction theory on this stack and a virtual fundamental class in its Chow group. Thus, our construction proves a conjecture of Sir Simon Donaldson on the existence of a virtual fundamental class for KSBA moduli spaces. A tautological invariant is defined by integrating a power of the first Chern class of the CM line bundle over the virtual fundamental class. This serves as a generalization of the tautological invariants defined by integrating tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.

Theorems & Definitions (145)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Definition 3.1