A Vanishing Theorem for Seiberg-witten Invariants

It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions have vanishing Seiberg-Witten invariants. Various vanishing theorems have played important roles in gauge theory. The first among them, due to S. K. Donaldson [2], states that the Donaldson invariants vanish for any smooth closed oriented 4-manifold X which decomposes to a connected sum X1#X2 with b2 (X1) > 0, b + 2 (X2) > 0. (Here b + 2 (Xi) is the dimension of a maximal subspace of H2(Xi,Z) on which the intersection pairing is positive definite.) E. Witten [12] has shown more recently that such a connected-sum manifold has also vanishing Seiberg-Witten invariants. Even though its proof is simple, Witten’s vanishing theorem is equally useful as Donaldson’s vanishing theorem; for example, it implies that any symplectic 4-manifold cannot be decomposed into the above connected sum by combining with Taubes’ theorem in [9]. In this note we show that Seiberg-Witten invariants vanish for another class of 4manifolds. These manifolds are obtained in connection with real algebraic geometry, and the construction was originally proposed in Donaldson [3]. See Remark 4 below. To state our theorem, recall that a map σ between two almost complex manifolds is called anti-holomorphic if σ∗J1 = −J2σ∗ on the tangent bundles, where Ji are the almost complex structures of the manifolds. In the following, K denotes the canonical bundle of an almost complex manifold (underlying a Kähler manifold). Theorem 1. Let X̃ be a Kähler surface with Ke X > 0 and b2 (X̃) > 3. Suppose that σ : X̃ → X̃ is an anti-holomorphic involution without fixed points. Then the quotient manifold X = X̃/σ has vanishing Seiberg-Witten invariants. In view of Witten’s vanishing theorem, it is natural to examine whether the quotient X can be decomposed into a connected sum: Proposition 2. If X̃ is a simply-connected Kähler surface or more generally symplectic 4-manifold with b2 (X̃) > 1 and σ : X̃ → X̃ is a free involution, then the quotient manifold X = X̃/σ is not diffeomorphic to any connected sum X1#X2 with both b2 (Xi) > 0. 1991 Mathematics Subject Classification. Primary 57R55, 57R57, 57N13; Secondary 14P25. Work supported by the Research Board grant of University of Missouri. Typeset by AMS-TEX 1