Chern classes of symplectic manifolds

Question

I have seen that people assign chern classes to the tangent bundle of symplectic manifolds. This confuses me, because to my knowledge chern classes detect differences in the complex structures of vector bundles.

I know that there is a canonical way to assign almost complex structures $J$ to symplectic manifolds $(M,\omega)$. However, this mechnism seems to depend on a choice of metric $g$.

(This is because locally there exists a matrix $A$ such that $\omega(v,w)=g(Av,w)$ and we can define a complex structure $J=Q^{-1} A$ where $Q^2=-A^2$.)

So why is this well defined?

Answer 1

The inclusion $U(n) \to Sp(2n,\mathbb R)$ is a maximal compact subgroup, hence a homotopy equivalence. This means principal $U(n)$-bundles are equivalent to principal $Sp(2n,\mathbb R)$-bundles. (Here is a reference, also see nLab page for symplectic group.)

As for your question about the metric - any two compatible almost complex structures lead to isomorphic complex vector bundles, because the space of CACS is path-connected, while the space of topological vector bundles on a manifold is discrete.

In more detail, given two CACS $J_0$ and $J_1$, you can build a path of almost complex structures $J_t$ (cf. Auroux's notes for path-connectedness of space of CACS), which leads to a structure of complex vector bundle on $p_M^*(TM) \to M \times I$. Then use the fact that any family of (topological) vector bundles over a (paracompact) manifold is constant (cf. Prop 1.7 of Hatcher's K-theory).

Answer 2

This is equivalent to Ben's answer but let me write me thoughts (which is most likely a rephrasing).

The point is that given a symplectic form $\omega$ on your manifold, the space of tamed/compatible almost complex structures is $\textbf{contractible}$ and, in particular, path connected. Thus any such complex structure will give you the same homotopic invariants and, in particular, the same chern classes.

This is a usual phenomenon when you want to construct topological invariants in differential geometry. Think of of the Chern-Weil construction for the Chern classes. You have to choose various auxiliary data (like a metric) for your bundle and you use this data to construct your class $\textbf{but}$ the crucial step is showing that the topological information you get is independent of the choices you made.