Symplectic and almost-complex curve data

A symplectic manifold is a smooth manifold $M$ with a smooth two-form $\omega$ that is closed and pointwise nondegenerate: $d\omega =0$, and ${\omega }_p^{\mathrm{♭}}:T_{p}M\to T_p^{\ast }M$, $v\mapsto {\omega }_p(v,\cdot )$, is an isomorphism at each $p$. Write $\dim M=2n$.

An almost-complex structure on $M$ is a smooth bundle endomorphism $J:TM\to TM$ with $J^2=-{\mathrm{id}}_{TM}$. It is $\omega$-tame if ${\omega }_p(v,J_{p}v)>0$ for every $p$ and every nonzero $v\in T_{p}M$. It is $\omega$-compatible if $g_J(v,w)=\omega (v,Jw)$ is a symmetric positive-definite inner product, as in the referenced definition. Compatibility implies ${\omega }_p(J_{p}v,J_{p}w)={\omega }_p(v,w)$. For compatible $J$, $g_J$ is a smooth Riemannian metric. For tame $J$, define the smooth symmetric form $g_J^{\mathrm{s}\mathrm{y}\mathrm{m}}(v,w)=\frac{1}{2}(\omega (v,Jw)+\omega (w,Jv))$.

A smooth Riemann surface $(\mathrm{\Sigma },j)$ is a smooth real two-manifold without boundary equipped with a holomorphic coordinate atlas: each chart ${\phi }_{\alpha }:U_{\alpha }\to V_{\alpha }\subset \mathbb{C}$ is a smooth coordinate chart, each transition ${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}$ is holomorphic on its domain, and $j$ is the tensor field given in these coordinates by $d{\phi }_{\alpha }^{-1}\circ i\circ d{\phi }_{\alpha }$. Thus “integrable” means precisely that $j$ arises from such an atlas. Its orientation is the one for which $(v,jv)$ is positive. For a smooth map $u:\mathrm{\Sigma }\to M$, say $u$ is $J$-holomorphic when $du\circ j=J\circ du$.

For such a map define its symplectic area by $E_{\omega }(u)={\int }_{\mathrm{\Sigma }}{u}^{\ast }\omega$ whenever the integral exists, and call it finite-area when this value is finite. This definition concerns smooth domains; nodal disk domains are defined separately.

A compatible almost-complex structure exists on every symplectic manifold under the source theorem's hypothesis ${\mathrm{A}\mathrm{C}}_{\omega }$, the axiom of countable choice. No integrability of $J$ is asserted.

Symplectic form and symplectic manifold

Compatible complex structure on a symplectic vector space

Every symplectic manifold admits a compatible almost-complex structure