Let $X$ be a variety. A symplectic structure on a holomorphic manifold/algebraic variety is a nondegenerate 2-form $\omega \in \Gamma(X, \Omega^2_{X})$ (where $\Omega^2_X$ is the sheaf of holomorphic or algebraic 2-forms on $X$) satisfying $d\omega = 0$.

Example: Let $M$ be a manifold, $X = T^*M$ its cotangent bundle. We construct a symplectic form $\omega$ as $d\lambda$ for some $\lambda : TX \to \mathbb{C}$, which ensures $d\omega = 0$.
Let $x \in M$ and $\alpha \in X_x$, the fibre over $x$. The projection map $\pi : X \to M$ induces a tangent map $\pi_{*} : T_\alpha X \to T_x M$ , and we define $\lambda$ by setting

In this description we can see naturality of $\lambda$, but for nondegeneracy and understanding how to evaluate the form an expression in coordinates is preferable. Let $q_1, \dots, q_{n}$ be local coordinates on $M$, and let $p_1, \dots, p_{n}$ be the corresponding dual coordinates on $T^*M$, so that $p_{i}\left( \frac{\partial}{\partial q_{i}} \right) = \delta_{ij}$ and $(q_{1}, \dots, q_{n}, p_{1}, \dots, p_{n})$ gives coordinates for $X$. Then elements of the tangent space $TX$ have the form

with $b_i, c_i \in \mathbb{C}$ and $(x, \alpha) \in X$ (so $x \in M, \alpha \in T^*_xM$). The map $\pi$ projects onto the $q$-coordinates, so the corresponding tangent map does the same:

Now we can calculate

using our dual coordinates, which identifies $\lambda$ with $\sum_{i} p_{i} \ \mathrm{d} q_{i}$ and hence $\omega$ with $\sum_{i} \mathrm{d}p_{i} \wedge \mathrm{d}q_{i}.$ Now $X$, being a holomorphic manifold, is locally isomorphic to $\mathbb{C}^{2n}$ and the local form of $\omega$ is nondegenerate here:

in the standard basis for $T^*\mathbb{C}^{2n}$, giving block matrix form

Hence we have a symplectic form on $T^*M$ for for any holomorphic manifold, agreeing in local coordinates with the canonical symplectic form on $\mathbb{C}^{2n}$.