ncts

In this talk, we discuss counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce two types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize polynomial Bridgeland stable objects in the bounded derived category. We construct reduced Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. We show that the cosections can be enhanced to (-1)-shifted closed 1-forms by generalizing the integration map of Pantev-Toen-Vaquie-Vezzosi which yields reduced (-2)-shifted symplectic derived enhancements on the moduli spaces. 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. This is joint work with Younghan Bae and Martijn Kool.