Virasoro conjecture for the stable pairs descendent theory of simply connected 3-folds (with applications to the Hilbert scheme of points of a surface)

TL;DR

This paper extends the Virasoro conjecture for stable pairs on 3-folds to cases with non-$(p,p)$-cohomology and proves it in special cases, linking it to Hilbert schemes of points and Fano varieties.

Contribution

It generalizes the Virasoro conjecture to broader 3-folds and proves it for specific cases involving Hilbert schemes and Fano varieties.

Findings

Virasoro constraints formulated for Hilbert schemes of points.

Universal formulas for descendents on $S^{[n]}$ established.

Full stable pairs theory computed for a cubic 3-fold.

Abstract

This paper concerns the recent Virasoro conjecture for the theory of stable pairs on a 3-fold proposed by Oblomkov, Okounkov, Pandharipande and the author in arXiv:2008.12514. Here we extend the conjecture to 3-folds with non-$(p,p)$-cohomology and we prove it in two specializations. For the first specialization, we let $S$ be a simply-connected surface and consider the moduli space $P_n(S\times \mathbb{P}^1, n[\mathbb{P}^1])$, which happens to be isomorphic to the Hilbert scheme $S^{[n]}$ of $n$ points on $S$. The Virasoro constraints for stable pairs, in this case, can be formulated entirely in terms of descendents in the Hilbert scheme of points. The two main ingredients of the proof are the toric case and the existence of universal formulas for integrals of descendents on $S^{[n]}$. The second specialization consists in taking the 3-fold $X$ to be a cubic and the curve class $\beta$ to be the line class. In this case we compute the full theory of stable pairs using the geometry of the Fano variety of lines.