The Seiberg–witten Invariants of Manifolds with Wells of Negative Curvature

A 4-manifold with b+ > 1 and a nonvanishing Seiberg–Witten invariant cannot admit a metric of positive scalar curvature. This remarkable fact is proved [18] using the Weitzenböck–Lichnerowicz formula for the square of the Spin Dirac operator, combined with the ‘curvature’ part of the Seiberg–Witten equations. Thus, in dimension 4, there is a strong generalization of Lichnerowicz’s vanishing theorem [8, 7, 12] for the index of the Dirac operator of a spin manifold with a metric of positive scalar curvature. In the recent years, the method of semigroup domination [3, 14, 13] has led to a different sort of generalization of Lichnerowicz’s theorem and other theorems in which a positive curvature hypothesis leads to a topological vanishing theorem. Essentially, the hypothesis of positive curvature may be weakened to permit negative curvature on a ‘small’ set. (The precise notion of ‘small’ depends on what kind of curvature is being discussed–see the statement of Theorem 1 for the version we are using.) In this note, we use semigroup domination to show that a 4-manifold with positive scalar curvature away from a set of small volume must have vanishing Seiberg–Witten invariants. Moreover, the same vanishing holds for the Seiberg–Witten invariant of any finite covering space. We sketch very briefly the definition of the Seiberg–Witten invariants, and refer to [10, 11, 9] for more details. Recall that a Spin structure σ on a smooth Riemannian 4-manifold X determines a pair of spinor bundles W → X which are Hermitian bundles over X of rank 2. A unitary connection A on L = det(W) determines the Dirac operator D A : Γ(W ) ∇A −−→ Γ(T X ⊗W) ρ −→Γ(W) where ∇A is the induced connection on W + and ρ denotes Clifford multiplication. The Seiberg–Witten equations, for a connection A and spinor φ are