Cyclic Group Actions on 4-Manifolds |
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Authors: | Yong Seung Cho Yoon Hi Hong |
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Institution: | (1) Department Of Mathematics, Ewha Women"s University, Seoul, 120-750, Korea |
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Abstract: | Let X be a closed, oriented Riemannian 4-manifold. Suppose that a cyclic group Z(
p
(p is prime) acts on X by an orientation preserving isometry with an embedded Riemann surface as fixed point set. We study the representation of Z
p
on the Spinc-bundles and the Z
p-invariant moduli space of the solutions of the Seiberg–Witten equations for a Spinc-structure X. When the Z
p
action on the determinant bundle det L acts non-trivially on the restriction L| over the fixed point set , we consider -twisted solutions of the Seiberg-Witten equations over a Spinc-structure ' on the quotient manifold X/Z
p
X', (0,1). We relate the Z
p
-invariant moduli space for the Spinc-structure on X and the -twisted moduli space for the Spinc-structure on X'. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the -twisted moduli space. When Z
p
acts trivially on L|, we prove that there is a one-to-one correspondence between the Z
p
-invariant moduli space M(
Zp
and the moduli space M (") where ' is a Spinc-structure on X' associated to the quotient bundle L/Z
p
X'. vskip0pt When p = 2, we apply the above constructions to a Kahler surface X with b
2
+
(X) > 3 and H
2(X;Z) has no 2-torsion on which an anti-holomorphic involution acts with fixed point set , a Lagrangian surface with genus greater than 0 and ]2H
2(H ;Z). If
K
X
2 > 0 or K
X
2
= 0 and the genus g()> 1, we have a vanishing theorem for Seiberg–Witten invariant of the quotient manifold X'. When K
X
2
= 0 and the genus g()= 1, if there is a Z
2-equivariant Spinc-structure on X whose virtual dimension of the Seiberg–Witten moduli space is zero then there is a Spinc-structure " on X' such that the Seiberg-Witten invariant is ±1. |
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Keywords: | cyclic group Spinc-structure Seiberg– Witten invariant holonomy quotient manifold Kä hler surface anti-holomorphic involution |
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