Holomorphic symplectic manifold I have two basic questions regarding hyperkaehler manifolds.

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*A holomorphic symplectic manifold is a complex manifold $X$ endowed with a $(2,0)$-form $\omega$.
I know that a Hyperkaehler manifold can be seen as a holomorphic symplectic manifold. But is the converse also true? Is every holomorphic symplectic manifold always Hyperkaehler? If not, can you give a counterexample?


*Does a Hyperkaehler manifold possess a natural polarization? What is it?
Thanks in advance for your answers.
 A: 1) The converse is not true. There has to be some compatibility between the complex structure and the symplectic structure. There are manifolds that admit both complex structures and symplectic structures, but without any such pair of structures being compatible. An example is the Kodaira-Thurston manifold.
Let $K=T^3 \times \mathbb{R}/\mathbb{Z}$ where $j \in \mathbb{Z}$ acts by
$$\psi: (x,y,z,t) \mapsto (x,y + jx, z, t+j).$$
The symplectic form $dx \wedge dy + dz \wedge dt$ is invariant under $\psi$.
The projection $\pi : K \to T^2$ given by $(x,y,z,t) \mapsto (z,t)$ has fibers that are symplectic $T^2$ with symplectic form the restriction of $dx \wedge dy$.
The universal cover is $\mathbb{R}^4$ with deck group
$$G = \{ (j_1,j_2,k_1,k_2) | \mbox{ in } \mathbb{Z}^4 , (\mathbb{j}',\mathbb{k}') * (\mathbb{j},\mathbb{k}) = (\mathbb{j}+\mathbb{j}',A_{\mathbb{j}'} \mathbb{k} + \mathbb{k}')\}$$
where $\mathbb{j}=(j_1,j_2)$ and $A_{\mathbb{j}}$ is the matrix
$$A_{\mathbb{j}} = \begin{pmatrix} 1 & j_2 \\ 0 & 1 \end{pmatrix}$$
So, $\pi_1(K)=G$.
Note that the subgroup $H=\langle (j_1,0,0,0) \rangle \cong \mathbb{Z} \leq G$ is a normal subgroup, and is the smallest normal subgroup such that the quotient $G/H$ is abelian, thence $H_1(K)$, which is the abelianization of $\pi_1(K)$ , is equal to $\mathbb{Z}^3$. Hence, the first Betti number is 3. But the first Betti number of a compact Kaehler manifold must be even, hence the manifold $K$ is not Kaehler and, in particular, not hyper-Kaehler.
2) Any Kaehler manifold $M$ (and therefore, any hyper-Kaehler manifold) has a polarization given by: $\mathcal{P}:=T_{(1,0)}(M)=\{ (x,v) \in T_x(M)^{\mathbb{C}} | J_x(v)=iv \}$, where $J : TM^{\mathbb{C}} \to TM^{\mathbb{C}}$ is the integrable almost complex structure induced by the complex structure on $M$. 
