A classification of Lagrangian planes in holomorphic symplectic varieties

Abstract

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\in H_2(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$-plane ${\mathbb{P}}^n\subset M$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $(R,R)=-\frac{n+3}{2}$ and the primitive such classes are all contained in a single monodromy orbit.

18 pages, comments welcome. v3: classification extended to all curve classes; some examples added. v4: to appear in J. Inst. Math. Jussieu

References in corpus (2)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

Cited by in corpus (2)

• Character Formulas on Cohomology of Deformations of Hilbert Schemes of K3 Surfaces
• An elementary description of nef cone for irreducible holomorphic symplectic manifolds