leaves, maximal symplectic submanifolds. The tangent space of the symplectic leaf at the point $p$ is $H(p)$.

### The Main Theorem

Corollary \[co:acyclic smooth\] implies that $X$ is smooth, hence it has the structure of a complex manifold. We have the following result.

\[th:symplectic leaves\] Let $\AA$ be an acyclic cluster algebra as defined above, $X$ an affine variety such that $\AA=\CC[X]$ and let $\Lambda$ define a compatible Poisson bracket. Then, $X$ is a holomorphic symplectic manifold.

First, let $p\in X$ be a generic point, by which we mean that $x_i(p)\ne 0$ for all $i=[1,\ldots,n]$. Set $x_i(p)=p_i$. It is easy to see that the Hamiltonian vectorfield $X_{x_i}$ at $p$ evaluates in the local coordinates $(x_1,\ldots, x_n)$ as $$X_{x_i}(p)= (\lambda_{1i} p_i,\ldots, \lambda_{ni} p_i)\ ,$$ where $\Lambda=(\lambda_{ij})_{i,j=1}^n$.

Since, $\Lambda$ is non-degenerate, we obtain that the Hamiltonian vectorfields span the tangent space $T_p X$ at $p$. It remains to consider the case when $p_i=0$ for some $i\in[1,n]$. Suppose that $p\in X$ such that $p_i=0$ and $p_j\ne 0$ for all $j<i$. We have to show that the symplectic leaf containing $p$ is not contained in the hyper-surface $x_i=0$. We may assume, employing induction, that if $p_1,\ldots, p_i\ne 0$ then the symplectic leaf at $p$ 