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Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$. Assume that $\omega$ is an arbitrary symplectic structure on $U$.

Is there a Riemannian metric on $U$ with the following property?

For every smooth function $f: U \to \mathbb{R}$, we have $J \nabla f = X_f$ where $X_f$ is the Hamiltonian vector field corresponding to $f$

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You are requiring that

$$g(\nabla f, X) = \omega (J\nabla f, X)$$

for all $f$ and for all $X$. So if $g$ exists, $g$ is of the form

$$g(X, Y) = \omega (JX, Y).$$

But the right hand side is not positive definite in general (for example $\omega = -\sum dx^i \wedge dy^i$).

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No. The two structures must be "compatible"; formally, this requires that:

$\omega(JX, JY) = g(X, JY) = g(JY, X) = \omega(J^2Y, X) = \omega(X, Y)$.

And if this is satisfied, then $g(X, Y) = \omega(JX, Y)$ is a not-necessarily-positive inner product.

A counterexample: consider the symplectic form $dx_1 \wedge dx_2 - dy_1 \wedge dy_2$ on $\mathbb{C}^2$.

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