Variation of moduli spaces and Donaldson invariants under change of polarization

TL;DR

This paper analyzes how moduli spaces of rank 2 sheaves and Donaldson invariants on surfaces with p_g=0 change when the polarization varies, providing explicit calculations of the lowest order terms of this change.

Contribution

It offers a detailed description of the polarization-dependent variation of moduli spaces and Donaldson invariants, including explicit computations of the first six terms of the change.

Findings

Explicit formulas for the change in moduli spaces under polarization variation.

Calculation of the first six terms of the change in Donaldson invariants.

Use of cohomology of Hilbert schemes for explicit computations.

Abstract

The paper determines the change of moduli spaces of rank $2$ sheaves on surfaces with $p_g=0$ under change of polarization and the corresponding change of the Donaldson invariants. In this revised version we have made some minor stylistic changes in the previous text. In addition we have added a final chapter of about 20 pages (announced in the previous version), in which the six lowest order terms (three of them non-zero) of the change are computed explicitely using computations in the cohomology of Hilbert schemes of points.