Almost Complex Structures

Almost Complex Structures on Symplectic Manifolds

Complex Structure

Let $V$ be a vector space. A complex structure on $V$ is a linear map: $J:V\to V\text{with}{J}^2=-Id.$ The pair $(V,J)$ is called a complex vector space.

Compatible Complex Structure

Let $(V,\Omega )$ be a symplectic vector space. A complex structure $J$ on $V$ is said to be compatible (with $\Omega$, or $\Omega$-compatible) if $G_J(u,v):=\Omega (u,Jv),\forall u,v\in V,\text{ is a positive inner product on }V.$ That is, $J\text{ is }\Omega \text{-compatible}⟺\{\begin{array}{ll}\Omega (Ju,Jv)=\Omega (u,v)& \text{symplectomorphism}\\ \Omega (u,Ju)>0,\forall u\ne 0& \text{taming condition}\end{array}$

Proposition

Let $(V,\Omega )$ be a symplectic vector space. Then there is a compatible complex structure $J$ on $V$

Almost Complex Structure

An almost complex structure on a smooth manifold $M$ is a smooth bundle map $J:TM\to TM$ such that $J^2=-\text{id}$. The pair $(M,J)$ is called an almost complex manifold.

Compatible Almost Complex Structures

Let $(M,\omega )$ be a symplectic manifold. An almost complex structure $J$ on $M$ is said to be compatible (with $\omega$, or $\omega$-compatible) if $\omega (Ju,Jv)=\omega (u,v)$ and $\omega (u,Ju)>0$ for all non-zero $u\in TM$. That is $g(u,v):=\omega (u,Jv)$ admits a Riemannian metric on $M$.

Proposition

Let $(M,\omega )$ be a symplectic manifold, and $g$ a riemannian metric on M. Then there exists a canonical almost complex structure $J$ on $M$ which is compatible.

Corollary

Any symplectic manifold has compatible almost complex structures.

Proposition

Let $(M,\omega )$ be a symplectic manifold, and $J_0,J_1$ two almost complex structures compatible with $\omega$. Then there is a smooth family $J_t$, $0\le t\le 1$, of compatible almost complex structures joining $J_0$ to $J_1$

Corollary

The set of all compatible almost complex structures on a symplectic manifold is path-connected.

Let $\mathcal{J}(T_{x}M,{\omega }_x)$ be the set of all compatible complex structures on $(T_{x}M,{\omega }_x)$ for $x\in M$.

Proposition

The set $\mathcal{J}(T_{x}M,{\omega }_x)$ is contractible, i.e., there exists a homotopy $h_t:\mathcal{J}(T_{x}M,{\omega }_x)⟶\mathcal{J}(T_{x}M,{\omega }_x)\text{ ,}0\le t\le 1\text{ ,}$ starting at the identity $h_0=\text{Id}$, finishing at a trivial map $h_1:\mathcal{J}(T_{x}M,{\omega }_x)\to \{J_0\}$, and fixing $J_0$ (i.e., $h_t(J_0)=J_0,\forall t$) for some $J_0\in \mathcal{J}(T_{x}M,{\omega }_x)$.

Sketch of Proof Fix a lagrangian subspace $L_0$ of $(V,\Omega )$. Let $\mathcal{L}(V,\Omega ,L_0)$ be the space of all lagrangian subspaces of $(V,\Omega )$ which intersect $L_0$ transversally. Let $\mathcal{G}(L_0)$ be the space of all positive inner products on $L_0$. Consider the map $\begin{array}{rl}\Psi :\mathcal{J}(V,\Omega )& \to \mathcal{L}(V,\Omega ,L_0)\times \mathcal{G}(L_0)\\ J& \mapsto (J{L}_0,G_J{|}_{L_0})\end{array}.$ It can be shown the map is a homeomorphism and $\mathcal{L}(V,\Omega ,L_0)$, $\mathcal{G}(L_0)$ are all contractible, thus $\mathcal{J}(V,\Omega )$ is followed as contractible. For surjectivity of $\Psi$, given $(L,G)\in \mathcal{L}(V,\Omega ,L_0)\times \mathcal{G}(L_0)$, define $J$ in the following manner: For $v\in L_0,v^{\perp }=\{u\in L_0\mid G(u,v)=0\}$ is a $(n-1)$-dimensional space of $L_0$; its symplectic orthogonal $(v^{\perp }{)}^{\Omega }$ is $(n+1)$-dimensional. Check that $(v^{\perp }{)}^{\Omega }\cap L$ is 1-dimensional. Let $Jv$ be the unique vector in this line such that $\Omega (v,Jv)=1$.Check that, if we take $v$‘s in some $G$-orthonormal basis of $L_0$, this defines the required element of $\mathcal{J}(V,\Omega )$.

Almost Complex Submanifold

A submanifold $X$ of an almost complex manifold $(M,J)$ is an almost complex submanifold when $J(TX)\subseteq TX$, i.e., for all $x\in X,v\in T_{x}X$, we have $J_{x}v\in T_{x}X$.

Proposition

Let $(M,\omega )$ be a symplectic manifold equipped with a compatible almost complex structure $J$. Then any almost complex submanifold $X$ of $(M,J)$ is a symplectic submanifold of $(M,\omega )$.

Remark

$S^2$ is an almost complex manifold and it is a complex manifold. $S^4$ is not an almost complex manifold (proved by Ehresmann and Hopf). $S^6$ is almost complex and it is not yet known whether it is complex. $S^8$ and higher spheres are not almost complex manifolds.