Seoul National University

Enumerative  invariants since 1995 are defined as integrals of cohomology classes over a particular homology class, called the virtual fundamental class. When there is a torus action, the virtual fundamental class is localized to the fixed points and this turned out to be the most effective technique for computation of the virtual integrals so far. About 10 years ago, Jun Li and I discovered that when there is a cosection of the obstruction sheaf, the virtual fundamental class is localized to the zero locus of the cosection. This also turned out to be quite useful for computation of Gromov-Witten invariants and more. In this talk, I will discuss a generalization of the cosection localization to real classes which provides us with a purely topological theory of Fan-Jarvis-Ruan-Witten invariants (quantum singularity theory) as well as some GLSM invariants. Based on a joint work with Jun Li at arXiv:1806.00116.
 

The space of smooth rational curves of degree d in projective space admits various moduli theoretic compactifications via GIT, stable maps, stable sheaves, Hilbert scheme and so on. I will discuss how these compactifications are related by explicit blow-ups and -downs for d