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We address the problem of classification of hyper-K\"ahler fourfolds with$b_2=23$. In particular we prove some special cases of the Conjecture ofO'Grady about hyper-K\"ahler $4$-folds numerically equivalent to the Hilbertscheme of two points on a $K3$ surface.

On irreducible symplectic 4-folds numerically equivalent to $(K3)^{[2]}$