A note on invariant pseudoholomorphic curves
Bull. Korean Math. Soc. 2001 Vol. 38, No. 2, 347-355
Yong Seung Cho and Do Sang Joe
Ewha Women's University, Ewha Women's University
Abstract : Let $(X,\omega)$ be a closed symplectic 4-manifold. Let a finite cyclic group $G$ act semifreely, holomorphically on $X$ as isometries with fixed point set $\Sigma$ (may be empty) which is a 2-dimension submanifold. Then there is a smooth structure on the quotient $X'=X/G$ such that the projection $\pi: X\to X'$ is a Lipschitz map. Let $L\to X$ be the $Spin^c$-structure on $X$ pulled back from a $Spin^c$-structure $L'\to X'$ and $b_2^+(X')>1$. If the Seiberg-Witten invariant $SW(L')\neq 0$ of $L'$ is non-zero and $L=E\otimes K^{-1}\otimes E$, then there is a $G$-invariant pseudo-holomorphic curve $u: C\to X$ such that the image $u(C)$ represents the fundamental class of the Poincar\'e dual $c_1(E)$. This is an equivariant version of the Taubes' Theorem.
Keywords : cyclic group action, pseudoholomorphic curve, Seiberg-Witten invariant
MSC numbers : 58G10, 57N13
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