Title: Intersection theory over relative Hilbert schemes and proof of S-duality conjecture for hyperplane sections in quintic threefold
Speaker: Artan Sheshmani (OSU)
Seminar URL: https://research.math.osu.edu/agseminar/
Abstract: I will talk about joint work with Amin Gholampour and
Richard Thomas on proving the S-duality conjecture regarding
modularity of DT invariants of sheaves with 2 diemnsional support
in an ambient CY threefold. One of the crucial ingredients needed
for our analysis is the relative Hilbert scheme of points on a
surface. More precisely, together with Gholampour we have proven
that the generating series, associated to the Hibert scheme of
points, relative to an effective divisor, on a smooth
quasi-projective surface is a modular form. This is a
generalization of the result of Okounkov-Carlsson for absolute
Hilbert schemes. We extend their constructions to the relative
setting, and using localization and degeneration techniques,
express the intersection numbers of the relative Hilbert scheme
in terms of tangent bundle of the surface with logarithmic zeros
and derive a nice formula as a modular form. I will then show how
this leads to the proof of a well known conjecture in string
theory, called S-duality modularity conjecture.
