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SEIBERG-WITTEN INVARIANTS AND (ANTI-)SYMPLECTIC INVOLUTIONS

Published online by Cambridge University Press:  10 September 2003

YONG SEUNG CHO
Affiliation:
Department of Mathematics, Ewha Women's University, Seoul 120-750, Korea e-mail: yescho@ewha.ac.kr
YOON HI HONG
Affiliation:
Department of Mathematics, Ewha Women's University, Seoul 120-750, Korea e-mail: yoonihong@hanmail.net
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Abstract

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Let $X$ be a closed, symplectic 4-manifold. Suppose that there is either a symplectic or an anti-symplectic involution $\sigma : X\,{\to}\, X$ with a 2-dimensional compact, oriented submanifold $\Sigma$ as a fixed point set.

If $\sigma$ is a symplectic involution then the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{\ge}\, 1$ is a symplectic 4-manifold.

If $\sigma$ is an anti-symplectic involution and $\Sigma$ has genus greater than 1 representing non-trivial homology class, we prove a vanishing theorem on Seiberg-Witten invariants of the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{ >}\,1.$

If $\Sigma$ is a torus with self-intersection number 0, we get a relation between the Seiberg-Witten invariants on $X$ and those of $X/\sigma$ with $b_2^+(X), b_2^+(X/\sigma)\,{ >}\,2$ which was obtained in [21] when the genus $g(\Sigma)\,{ >}\,1$ and $\Sigma\cdot\Sigma\,{=}\,0$.This work was supported by a Korea Research Foundation Grant (No KRF-2002-072-C00010).

Type
Research Article
Copyright
© 2003 Glasgow Mathematical Journal Trust