##### Department of Mathematics,

University of California San Diego

****************************

### Math 208 - Algebraic Geometry Seminar

## Arkadij Bojko

#### University of Oxford

## Wall-crossing for Hilbert schemes on fourfolds and Quot-schemes on surfaces

##### Abstract:

Virtual counting of coherent sheaves has seen recently a large development in complex dimension four, where it was defined for Calabi--Yau fourfolds by Borisov--Joyce and Oh--Thomas. I will focus on invariants for Hilbert schemes of points as they have not been well understood before. The only known result expressed integrals of top Chern classes of tautological vector bundles associated to smooth divisors in terms of the MacMahon function and Cao--Kool conjectured this holds for any line bundle. To address these questions I discuss the conjectural wall-crossing formulae of Joyce and discuss how to relate them to the conjectures on Hilbert schemes. On the other hand, Arbesfeld--Johnson--Lim--Oprea--Pandharipande studied Quot-schemes on surfaces and their virtual integrals giving explicit expressions for their generating series. Interestingly, these satisfy similar wall-crossing formulae as Hilbert schemes in the fourfold case when the curve class is zero. As a consequence their general invariants share a large similarity. Computing explicitly virtual fundamental classes and integrals on both, we can firstly recover the results in the five author paper from a small piece of data. Moreover, we obtain a universal transformation comparing integrals on Hilbert schemes on fourfolds and elliptic surfaces.

Host: Dragos Oprea

### April 9, 2021

### 12:00 PM

### Location / Remote Access URL: Contact David Stapleton dstapleton@ucsd.edu, for zoom access.

****************************