Virtual Segre and Verlinde numbers of projective surfaces

Abstract

Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces.Extending the work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg-Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño-Moore conjecture for higher rank Donaldson invariants. For ranks 3 and 4, we obtain explicit expressions for Donaldson invariants in terms of Seiberg-Witten invariants.

Tautological integrals on Hilbert scheme of points I

“…On surfaces Hilb k (S ) = GHilb k (S ), and the cohomological intersection theory of Hilb k (S ) can be approached from several different directions: 1) via the inductive recursions set up in [14,15,16]; 2) using Nakajima calculus [39,24,33,20] or more recently 3) virtual localisation on Quot schemes [22,35]. Lehn's conjecture [33] on top Segre numbers of tautological line bundles, and its recent extension, the Segre-Verlinde duality conjectures [36,29,21,37], 1 incapsulate the complexity of this theory. However, these techniques fail at a fundamental level in higher dimensions.…”

Cosection localization and the Quot scheme QuotSl(E)

“…By integrating universal cohomology classes over it, one obtains invariants of these moduli spaces. For (2), these are the (algebraic) Donaldson invariants of S, which are well studied, see, for instance, [1][2][3][4][5][6][7][8][9]; the study of the invariants corresponding to (1) is the subject matter of this paper. The schemes Quot l S (E) satisfy three basic properties:…”