Lothar Göttsche, ICTP
Title: Refined curve counting
Abstract: We define and study two closely related versions of refined curve counting invariants of algebraic surfaces: one is in terms of the χy −genus of the relative Hilbert schemes.
As relative Hilbert schemes are closely related to Pandharipande-Thomas moduli spaces, these could be viewed as refined BPS state counting.
The other version is via tropical geometry. These invariants are polynomials in a variable y that interpolate between the Severi degrees counting complex curves and the Welschinger invariants, which could real curves. Under suitable assumptions both versions of refined invariants are conjectured to coincide. We will study the generating functions for these invariants, and express them in terms of a Heisenberg algebra action. We also consider refined descendant invariants, that interpolate between more general Welschinger invariants and Gromov-Witten invariants with descendants.
