Symplectic Resolutions for Symmetric Products of Surfaces

Let $S$ be a smooth complex connected analytic surface which admits a holomorphic symplectic structure. Let $S^{(n)}$ be its $n$th symmetric product. We prove that every projective symplectic resolution of $S^{(n)}$ is isomorphic to the Douady-Barlet resolution $S^{[n]}\to S^{(n)}$.