Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type

We classify the cohomology classes of Lagrangian 4-planes ℙ 4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C , for which ( C, C ) ≥ 0, and nodal classes C , for which ( C, C ) = −2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that ( C, C ) = −γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C . In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = ℓ of a line in a smooth Lagrangian n -plane ℙ n must satisfy ( ℓ , ℓ ) = −( n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X , and showing there is a unique monodromy orbit of Lagrangian 4-planes.