EQUIDIMENSIONALITY OF LAGRANGIAN FIBRATIONS ON HOLOMORPHIC SYMPLECTIC MANIFOLDS

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Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type (2013, background): “All fibers of s have pure dimension n [29].”
Base manifolds for fibrations of projective irreducible symplectic manifolds (2007, unknown): “(1), (2) and (3) in Theorem 1.1 were proved in [Ma1] and [ Ma2 ].”
On the Beauville form of the known irreducible symplectic varieties (2006, background): “By the previous lemma V is dense in � −1 (R 0 ) and, since R 0 is dense in R and the fibers of � are equidimensional (see [ 13 ]), the open subvariety V is dense in the divisor � −1 (R).”
Twisted Stability and Fourier–Mukai Transform I (2001, unknown): “For a smooth curve C 2 Hilb x , the fiber of Msþkf ðw þ R X Þ! Hilb x is Pic rþ1 ðCÞ .I t is easy to see that Pic rþ1 ðCÞ is a Lagrangian subscheme of Msþkf ðw þ RXÞ .B y Matsushita [ Mt ], every fiber is of dimension ðx 2 Þ=2 þ 1. &”