[Paper Review] Variation of moduli spaces and Donaldson invariants under change of polarization

Geir Ellingsrud, Lothar Göttsche|ArXiv.org|Oct 7, 1994

Algebraic Geometry and Number Theory17 references19 citations

TL;DR

This paper studies the variation of moduli spaces of rank 2 torsion-free sheaves on algebraic surfaces under change of polarization, showing that wall-crossing induces a sequence of smooth blow-ups and blow-downs along projective bundles over products of Hilbert schemes of points. The key contribution is an explicit computation of the change in Donaldson invariants for surfaces with p_g = q = 0, expressed in terms of cohomology classes on Hilbert schemes, confirming the conjecture of Kotschick and Morgan for the lowest six terms with three vanishing terms.

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.

Motivation & Objective

• To understand how moduli spaces of rank 2 sheaves change when the polarization crosses a wall in the ample cone.
• To describe the geometric transformation of the moduli space as a sequence of smooth blow-ups and blow-downs when crossing a 'good' wall.
• To compute the change in Donaldson invariants under polarization change for surfaces with p_g = q = 0.
• To verify the conjecture of Kotschick and Morgan on the structure of the change in Donaldson invariants, particularly the vanishing of three of the lowest-order terms.

Proposed method

• Uses elementary transforms of universal sheaves to describe wall-crossing as a sequence of flips analogous to Mori's minimal model program.
• Models the change in moduli spaces as smooth blow-ups along projective bundles over products of Hilbert schemes of points on the surface.
• Applies cohomological computations on Hilbert schemes of points to express the change in Donaldson invariants in terms of natural classes.
• Employs symmetric product cohomology and equivariant integration techniques to compute intersection numbers on S^d.
• Derives a polynomial expression for the change in Donaldson invariants modulo certain monomials, using recursive formulas and symmetric functions.
• Verifies compatibility with the Kotschick-Morgan conjecture by computing the six lowest-order terms explicitly.

Experimental results

Research questions

• RQ1 How does the moduli space M_H(c1,c2) change when the polarization H crosses a wall in the ample cone of a surface S?
• RQ2 What is the geometric nature of the transformation between moduli spaces across a wall, and can it be described via blow-ups and blow-downs?
• RQ3 How do Donaldson invariants change under polarization variation, and can this change be computed explicitly for surfaces with p_g = q = 0?
• RQ4 Do the six lowest-order terms in the change of Donaldson invariants match the conjecture of Kotschick and Morgan, particularly the prediction that three of them vanish?
• RQ5 Can the change in invariants be expressed purely in terms of cohomology classes on Hilbert schemes of points on S?

Key findings

• The wall-crossing transformation of the moduli space is realized as a sequence of smooth blow-ups along projective bundles over products of Hilbert schemes of points, followed by smooth blow-downs.
• For K3 or abelian surfaces, the change is an elementary transformation of symplectic varieties, preserving the symplectic structure.
• The change in Donaldson invariants is computed explicitly in terms of natural cohomology classes on Hilbert schemes of points on S.
• The six lowest-order terms in the change of Donaldson invariants are computed, and three of them are found to vanish, confirming the Kotschick-Morgan conjecture.
• The results are compatible with the conjectured formula for the change, particularly the prediction that the terms of degree 2, 4, and 6 in the polynomial are zero.
• The computation relies on symmetric product cohomology and intersection theory on S^d, with key identities derived using equivariant integration and symmetric function identities.