$$X_a(b)=\{a,b\}\ .$$ This derivation is called the [*Hamiltonian vectorfield*]{} of $a$ on $A$.

Now suppose that $A$ is the coordinate ring of an affine complex variety $Y$. We will associate to the Poisson bracket the [*Poisson bivector*]{} $u\in \Lambda^2 T(Y)$ where $T(Y)$ denotes the tangent bundle of $Y$. Let $p\in Y$ be a point and $\mm_p\subset A$ the corresponding maximal ideal. Let $\alpha,\beta\in \mm_p/\mm_p^2$ be elements of the cotangent space and let $f,g$ be lifts of $\alpha$ and $\beta$, respectively. We define $u_p\in \Lambda^2 T_p(Y)$ $$u_p(\alpha,\beta)=\{f,g\}(p)\ .$$ Note that $u_p$ is a well-defined skew-symmetric form. Indeed, if $I\subset A$ is an ideal and $b\in \II^2$, then for all $a\in A$ $$\{a,b\}\in \II$$ by the Leibniz rule. The form $u_P$ may be degenerate, indeed if it is non-degenerate at every point $p\in Y$, then we call $u_P$ symplectic and, moreover, if $Y$ is connected, then $Y$ is smooth and a (holomorphic) symplectic manifold. Define $$N(p)=\{\alpha \in T^*_p Y:u_p(\alpha,\cdot)=0\}\ ,$$ and $H(p)\subset T_p Y$ its orthogonal complement. The space $H(P)$ is the tangent space of the linear span of the Hamiltonian vectorfields at $p$. Recall that by the Theorem of Frobenius, a Poisson variety $Y$ decomposes as a disjoint union of symplectic 