Pseudoholomorphic curves

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1 Introduction

In this article, $(X,J)$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; In this article, $(X,J)$ will denote an [http://en.wikipedia.org/wiki/Almost_complex_manifold almost complex manifold] of dimension n$. {{beginthm|Definition|($J$-holomorphic curve)}} Let $(\Sigma,j)$ be a Riemann surface with complex structure $j$. A '''$(j,J)$-holomorphic curve''' $$u\co\Sigma\rightarrow X$$ is a smooth map satisfying $du\circ j=J\circ du$. A $J$-holomorphic map is called '''simple''' if it cannot be factored as $u=v\circ\phi$ where $\phi$ is a $(j',j)$-holomorphic branched cover $\phi\co\Sigma'\rightarrow\Sigma$ of degree strictly greater than 1. We will usually omit $j$ from the notation and speak of $J$-holomorphic curves. The term '''pseudoholomorphic curve''' will be used to describe a $J$-holomorphic curve when we do not want to specify $J$. {{endthm}} Pseudoholomorphic curves provide a useful tool for studying [[symplectic manifolds]]. </wikitex> == References == {{#RefList:}} [[Category:Theory]](X,J)$ will denote an almost complex manifold of dimension $2n$$2n$.

Definition 1.1 ($J$$J$-holomorphic curve). Let $(\Sigma,j)$$(\Sigma,j)$ be a Riemann surface with complex structure $j$$j$. A $(j,J)$$(j,J)$-holomorphic curve

$\displaystyle u\co\Sigma\rightarrow X$ $$\displaystyle u\co\Sigma\rightarrow X$$

is a smooth map satisfying $du\circ j=J\circ du$$du\circ j=J\circ du$. A $J$$J$-holomorphic map is called simple if it cannot be factored as $u=v\circ\phi$$u=v\circ\phi$ where $\phi$$\phi$ is a $(j',j)$$(j',j)$-holomorphic branched cover $\phi\co\Sigma'\rightarrow\Sigma$$\phi\co\Sigma'\rightarrow\Sigma$ of degree strictly greater than 1. We will usually omit $j$$j$ from the notation and speak of $J$$J$-holomorphic curves. The term pseudoholomorphic curve will be used to describe a $J$$J$-holomorphic curve when we do not want to specify $J$$J$.

Pseudoholomorphic curves provide a useful tool for studying symplectic manifolds.