Must a Hermitian/Kähler Manifold have a complex structure?

Question

Recall the following definitions:

A Hermitian manifold is a smooth manifold $M$ endowed with a tensor field $J\in\mathcal{T}^1_1(M)$ and a Riemannian metric $g$ such that $$\forall x\in M,\ (J_x)^2 = -\mathrm{id}_{\mathrm{T}_x}$$ and such that the tensor field $\omega\in\mathcal{T}^2_0(M)$ defined by $\omega(-,-):=g(-,J(-))$ is a nondegenerate differential 2-form.

Additionally, a Hermitian manifold $(M,J,g,\omega)$ is said to be Kähler if $\omega$ is closed – i.e. if $(M,\omega)$ is a symplectic manifold.

Now, the tensor field $J$ allows us to consider $\mathrm{T}_x$ as a complex vector space for each $x\in M$, by defining, for $v\in\mathrm{T}_x$ and for $\lambda=\lambda_1 + i\lambda_2\in\mathbb{C}$, $$\lambda\, v = \lambda_1\, v + \lambda_2\, J(v).$$

My aim is to use this to show that in fact, for any smooth chart $(U,\varphi)$ of $M$ with $\varphi:U\hookrightarrow\mathbb{R}^{2n}$, we can identify $\mathbb{R}^{2n}$ with $\mathbb{C}^n$, and obtain a holomorphic atlas.

Is anything like this possible? Or does there exist a Kähler manifold that cannot be endowed with (holomorphic) complex structure?

Answer 1

Your definitions of hermitian and Kähler are not standard - the standard definitions assume that $J$ is integrable. The notions you define are usually called almost hermitian and almost Kähler respectively.

With this standard terminology, your question then becomes:

Is every almost Kähler manifold a complex manifold?

As you pointed out, any almost Kähler manifold is symplectic. Conversely, given a symplectic form $\omega$ and a Riemannian metric $g$, one can find an almost complex structure $J$ such that $\omega(X, Y) = g(JX, Y)$. Therefore, any symplectic manifold can be equipped with a compatible almost complex structure and hence viewed as an almost Kähler manifold.

So the question can again be reformulated as follows:

Is every symplectic manifold a complex manifold?

The answer is no. The first examples of compact symplectic manifolds which do not admit a complex structure were found by Fernández, Gotay, and Gray in this paper. They are principal circle bundles over the total space of a principal circle bundle over $S^1\times S^1$.

In fact, there are many counterexamples. Gompf showed (in this paper) that for any finitely presented group $G$ there is a compact four-manifold $M$ with $\pi_1(M) = G$ which is symplectic but not complex; in fact, $M$ is not even homotopy equivalent to a complex surface. This result follows from Theorem $6.2$.