Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4–manifolds

Abstract

In the first part of the paper we prove an existence theorem for gauge invariant $L^2$–normal neighborhoods of the reduction loci in the space ${\mathcal{A}}_a\left(E\right)$ of oriented connections on a fixed Hermitian 2–bundle $E$. We use this to obtain results on the topology of the moduli space ${\mathcal{ℬ}}_a\left(E\right)$ of (non-necessarily irreducible) oriented connections, and to study the Donaldson $\mu$–classes globally around the reduction loci. In this part of the article we use essentially the concept of harmonic section in a sphere bundle with respect to an Euclidean connection.

Second, we concentrate on moduli spaces of instantons on definite 4–manifolds with arbitrary first Betti number. We prove strong generic regularity results which imply (for bundles with “odd" first Chern class) the existence of a connected, dense open set of “good" metrics for which all the reductions in the Uhlenbeck compactification of the moduli space are simultaneously regular. These results can be used to define new Donaldson type invariants for definite 4–manifolds. The idea behind this construction is to notice that, for a good metric $g$, the geometry of the instanton moduli spaces around the reduction loci is always the same, independently of the choice of $g$. The connectedness of the space of good metrics is important, in order to prove that no wall-crossing phenomena (jumps of invariants) occur. Moreover, we notice that, for low instanton numbers, the corresponding moduli spaces are a priori compact and contain no reductions at all so, in these cases, the existence of well-defined Donaldson type invariants is obvious. Note that, on the other hand, there seems to be very difficult to introduce well defined numerical Seiberg–Witten invariants for definite 4–manifolds. For instance, the construction proposed by Okonek and the author in [Seiberg–Witten invariants for 4–manifolds with $b_{+}=0$, from: ”Complex analysis and algebraic geometry”, (T Peternell, F O Schreyer, editors), de Gruyter, Berlin (2000) 347–357] gives a $\mathbb{Z}$–valued function defined on a countable set of chambers.

The natural question is to decide whether these new Donaldson type invariants yield essentially new differential topological information on the base manifold, or have a purely topological nature.