Lagrangian hyperplanes in holomorphic symplectic varieties

Details Lagrangian hyperplanes in holomorphic symplectic varieties Lagrangian hyperplanes in holomorphic symplectic varieties

2011-10-31

arxiv.org/abs/1111.0047v2

Mathematics

Algebraic Geometry

Scientific paper

We classify the cohomology classes of Lagrangian hyperplanes $\mathbb{P}^4$ in a smooth manifold $X$ deformation equivalent to a Hilbert scheme of 4 points on a K3 surface, up to the monodromy action. Classically, the cone of effective curves on a K3 surface $S$ is generated by nonegative classes $C$, for which $(C,C)\geq0$, and nodal classes $C$, for which $(C,C)=-2$; Hassett and Tschinkel conjecture that the cone of effective curves on a holomorphic symplectic variety $X$ is similarly controlled by "nodal" classes $C$ such that $(C,C)=-\gamma$, for $(\cdot,\cdot)$ now the Beauville-Bogomolov form, where $\gamma$ 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 $C=\ell$ of a line in a smooth Lagrangian $n$-plane $\mathbb{P}^n$ must satisfy $(\ell,\ell)=-\frac{n+3}{2}$. 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 hyperplanes.

Affiliated with

Also associated with

No associations

Rating

Lagrangian hyperplanes in holomorphic symplectic varieties does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Lagrangian hyperplanes in holomorphic symplectic varieties, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lagrangian hyperplanes in holomorphic symplectic varieties will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-466677