Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2008; 45(5): 1427-1442

Printed September 1, 2008

Copyright © The Korean Mathematical Society.

Counting real $J$-holomorphic discs and spheres in dimension four and six

Cheol-Hyun Cho

Seoul National University

Abstract

We provide another proof that the signed count of the real $J$-holomorphic spheres (or $J$-holomorphic discs) passing through a generic real configuration of $k$ points is independent of the choice of the real configuration and the choice of $J$, if the dimension of the Lagrangian submanifold $L$ (fixed point set of involution) is two or three, and also if we assume $L$ is orientable and relatively spin. We also assume that $M$ is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the signed degree of an evaluation map.

Keywords: holomorphic discs, anti-symplectic involution, Welschinger invariants

MSC numbers: 54D45, 14N35