Donaldson Polynomials of Algebraic Surfaces

Abstract

Let S be an algebraic surface and C a smooth algebraic curve on S. In this chapter we shall give a recipe for computing the Donaldson polynomial γc (S) evaluated on C, based on the enumerative geometry of certain divisors in the moduli space M c ,(S, L) which represent the divisors µ(C).This recipe will be used in Chapter VII to make calculations for elliptic surfaces. The divisors representing µ(C) are supported on the set of vector bundles in the moduli space where certain sheaf cohomology groups are nonvanishing. The situation is thus identical to the discussion in Chapter III which identified geometric representatives for µ(C) in terms of the cohomology groups of an elliptic complex formed from the coupled Dirac operator instead of the sheaf cohomology of holomorphic vector bundles (i.e. the \( \bar \partial\)-operator). Indeed, as we shall see in Section 3, the Dirac operator may be identified with the \( \bar \partial\)-operator, so that the divisors described above are in fact the same divisors as those described in Chapter III. However, the viewpoint of algebraic geometry gives us some extra information. The sets in the moduli space described above are naturally divisors, that is complex hypersurfaces with integer multiplicities. As we shall see, algebraic geometry provides a natural method for calculating these multiplicities. Often, however, we could avoid some of the more technical arguments given in this chapter and make shorter ad hoc arguments, for instance by working purely within algebraic geometry or by evaluating the multiplicities by looking at compact 2-cycles contained in the (non-compactified) moduli space. Still, it seems worthwhile to have a very general framework for describing the multiplicities for future applications. Our main result is Theorem 1.9, which gives a general method for evaluating the Donaldson polynomial of an algebraic surface on cohomology classes represented by algebraic curves. For the applications in Chapter VII, we will need a more specialized variant of this result, Theorem 1.12. We also discuss analogous results when the moduli space has the expected dimension but is nonreduced. Section 2 is devoted to a proof of Donaldson’s theorem that the polynomial invariants do not all vanish for a simply connected algebraic surface. The proof draws on many of the ideas described in the first section and in Chapter IV. The last section is a somewhat technical appendix which defines and compares various notions of determinant line bundles.