Compatible almost-complex structure - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold. A almost-complex structure $J$ on $M$ is compatible with $\omega$ if

\omega(Jv,Jw)=\omega(v,w)

for all tangent vectors $v,w$ based at the same point, and

g_J(v,w)=\omega(v,Jw)

is positive definite. The first condition makes $g_J$ symmetric, so $g_J$ is a Riemannian metric. Compatibility is a pointwise condition varying smoothly with the base point; it is stronger than merely requiring $\omega(v,Jv)>0$ , which defines a tamed almost-complex structure.

Equivalent linear-algebra formulation

At each $p\in M$ , compatibility says that $J_p$ is a complex structure on the symplectic vector space $(T_pM,\omega_p)$ , that $J_p$ is symplectic, and that $\omega_p(\,\cdot\,,J_p\,\cdot\,)$ is an inner product. Equivalently, the three tensors satisfy

\omega(v,w)=g_J(Jv,w),\qquad g_J(Jv,Jw)=g_J(v,w).

Existence and deformation

Every symplectic vector bundle admits compatible complex structures. Consequently every symplectic manifold has compatible almost-complex structures, and the space of all such structures is nonempty and contractible. This flexibility is central to constructions whose final invariant should not depend on the auxiliary choice of $J$ ; see McDuff and Salamon, §2.5.

Relation to complex and Kähler geometry

Compatibility does not imply that $J$ comes from holomorphic coordinates. When $J$ is additionally integrable, the triple $(M,\omega,J)$ , with metric $g_J$ , is a Kähler manifold. Thus compatibility supplies the metric bridge between symplectic and complex geometry, while integrability is a separate differential condition.

References

D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology, 2nd ed., American Mathematical Society, 2012. AMS DOI record. Relevant: §2.5.
A. Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2001. Springer DOI record. Relevant: compatible complex structures on symplectic vector spaces and bundles.