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Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it is an irreducible symplectic variety, although it may fail to be projective.

Are there any criteria/known cases where one can guarantee the projectivity of $\widetilde{X}$?

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