Counting surfaces on Calabi-Yau 4-folds I: Foundations

TL;DR

This paper develops foundational tools for counting surfaces on Calabi-Yau 4-folds, introducing new moduli spaces, relating them via wall-crossing, and establishing invariance properties of their virtual cycles.

Contribution

It introduces two new types of stable pair moduli spaces, relates them to the Hilbert scheme, and constructs reduced virtual cycles with deformation invariance.

Findings

Established relations between moduli spaces via GIT wall-crossing

Constructed reduced Oh-Thomas virtual cycles

Proved deformation invariance along Hodge loci

Abstract

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category. We construct \emph{reduced} Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero reduced virtual cycle.