ON MODULI SPACES OF REAL CURVES IN SYMPLECTIC MANIFOLDS

Abstract

Given a symplectic manifold $(X,\omega)$, an almost complex structure $J$, and an antisymplectic involution $\phi$, we study genus zero real $J$-holomorphic curves in $X$. There are two types of such curves, those that can be divided into two $J$-holomorphic discs and those that cannot. Moduli spaces of $J$-holomorphic discs are more studied in the literature; in this case, we develop and use some degeneration techniques to add to the previous results and get a better understanding of these moduli spaces. We also study the second case, for which the orientation problem is different and define (and calculate) some invariants using these moduli spaces. As shown in this thesis, these two cases are tied together and often need to be combined to get a fully well-defined theory.