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

This is the first in a series of papers on counting surfaces on Calabi-Yau 4-folds.To complement 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 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.As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a nonzero reduced virtual cycle.1. Introduction 2431 2. Stable pairs 2443 3. Moduli spaces 2453 4. Virtual cycles 2481 5. Deformation invariance 2513 Appendix A. Reductions via .1/-shifted 1-forms 2529 Appendix B. Moduli stacks of semistable sheaves 2543 Appendix C. Orientations for relative moduli spaces 2554