The virtual K-theory of Quot schemes of surfaces

Abstract

We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces X with pg > 0. We also show that the generating series of virtual cobordism classes can be irrational.Given a K-theory class on X of rank r, we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases:(i) Quot schemes of dimension 0 quotients, (ii) Hilbert schemes of points and curves over surfaces with pg > 0, (iii) Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes. Moreover, for punctual quotients of the trivial sheaf of rank N , we prove a new symmetry of the Segre/Verlinde series exchanging r and N . The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.

Wall-crossing for punctual Quot-schemes

“…We study a new 12-fold correspondence relating invariants of Calabi-Yau fourfolds, surfaces and curves. This includes a correspondence between virtual Segre and Verlinde series and improves on the 8-fold correspondence observed in [1].4. We study the higher rank Nekrasov genus and its cohomological limit both of which can be expressed in terms of the Mac-Mahon series M (q) = n>0 (1 − q n ) −n .Along the way, we proved a new combinatorial identity related to Lagrange inversion which appeared in the companion paper [6].…”

“…We study a new 12-fold correspondence relating invariants of Calabi-Yau fourfolds, surfaces and curves. This includes a correspondence between virtual Segre and Verlinde series and improves on the 8-fold correspondence observed in [1].…”

Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi--Yau 4-folds

“…Proof. Arbesfeld-Johnson-Lim-Oprea-Pandharipande [2] prove general formulae for generating series n∈Z q n…”

“…Their virtual analogs were studied in [28] and [2]. Using this as an inspiration together with the relation to higher rank Nekrasov genus (see Remark 6.6), we define square root Verlinde series…”

“…We discuss now a stronger correlation between invariants on surfaces and CY 4-folds. To a reader familiar with the results of [60] and [2], this will be a generalization of most of §1.3 and §1.5.…”