Recall that given a symplectic manifold \((M,\omega)\), and a Hamiltonian function \(H:M\to \mathbb{R}\), the Hamiltonian vector field \(X_H\) is defined by
\[ \omega(\cdot,X_H) = dH \in Z^1(\Omega_M^\bullet, M). \]
On \(\mathbb{C}^2=\mathbb{C}e_x\oplus \mathbb{C}e_y\) with \(\omega = dx\wedge dy\), the Hamiltonian vector field of \(H\) is given by
\[ \omega(\cdot,X_H) = X_H^{(y)}dx - X_H^{(x)}dy = dH = \frac{\partial H}{\partial x}dx + \frac{\partial H}{\partial y}dy. \]
So
\[ X_H^{(x)} = -\frac{\partial H}{\partial y}, \quad X_H^{(y)} = \frac{\partial H}{\partial x}. \]
The Poisson structure on a symplectic manifold is defined by
\[ \{f,g\} = \omega(X_f,X_g) = X_f g = -X_g f. \]
On \((\mathbb{C}^2, dx\wedge dy)\), the Poisson bracket is given by
\[ \{,\}:\mathcal{O}\wedge \mathcal{O}\to \mathcal{O}:\quad \{f,g\} = \frac{\partial f}{\partial x}\frac{\partial g}{\partial y} - \frac{\partial f}{\partial y}\frac{\partial g}{\partial x}. \]
\[ \{x^2,\} = 2x\frac{\partial }{\partial y},\quad \{xy,\} = y\frac{\partial }{\partial y} - x\frac{\partial }{\partial x},\quad \{y^2,\} = -2y\frac{\partial }{\partial x}. \]