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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