Veranstaltungen

Motivated by strange duality, Johnson predicted a correspondence between the Segre and the Verlinde series in the case of a Hilbert scheme of points on a surface. This was soon after proved by Marian-Oprea-Pandharipande and motivated an analogous result in the case of punctual quot-schemes of trivial vector bundles on curves and surfaces. In my work, I have observed that this mysterious correspondence holds even after allowing more general vector bundles, and I extended it to Calabi-Yau fourfolds. Combining it with a natural-looking symmetry between Segre and Verlinde series, respectively, lead to a 12-fold correspondence of these invariants in 1,2, and 4 dimensions. However, very little was known about the equivariant versions of these results. In this talk, I will give an overview of our joint project with J. Huang which addresses multiple open questions by either giving complete theorems proving the correspondences or formulating conjectures supported by empirical evidence.

As a continuation of the previous lecture, this talk will outline the proofs of the theorems previously mentioned. I will mainly focus on surfaces and describe the proof of a universal series expression for the equivariant Segre and Verlinde invariants. This is done by relating them to their non-equivariant version of projective toric surfaces. In the case of K-trivial surfaces, a simpler expression can then be obtained for the reduced invariants. Using combinatorial tools introduced by L. Gottsche and A. Mellit, some of the universal series can be computed explicitly. I shall also display some computer calculations for Calabi-Yau 4-folds that support the conjectures previously mentioned. These statements can be considered as 4-D analogues to the results on surfaces.