### January 12, 2022

TITLE: Counting surfaces on Calabi-Yau fourfolds

ABSTRACT: On the sheaf side, there are two well-known ways to count curves on threefolds: via 1-dimensional subschemes (DT theory) and via stable pairs (PT theory). We show that for surfaces on fourfolds there are three theories: 2-dimensional subschemes and *two* types of stable pairs. Using Oh-Thomas/Borisov-Joyce, this allows us to define DT, PT0, PT1 invariants of Calabi-Yau fourfolds. We reduce the theory and prove that the resulting invariants are deformation invariant over the Hodge locus. This can be used to show the variational Hodge conjecture in some examples. We conjecture a DT-PT0 correspondence, which we check in non-compact examples using toric geometry and in compact examples using virtual pull-back. We conjecture a PT0-PT1 correspondence on Weierstrass elliptic fourfolds, which we prove for certain vertical classes. Joint work with Y. Bae and H. Park.