Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

TL;DR

This paper constructs rational curves on certain holomorphic symplectic manifolds using Brill-Noether theory, identifies wall divisors, and describes algebraically coisotropic subvarieties, supporting Voisin's conjecture on the Chow ring.

Contribution

It introduces a method to realize all wall divisors via duals to rational curves and describes the associated coisotropic subvarieties, advancing understanding of the birational geometry of these manifolds.

Findings

All wall divisors can be obtained as duals to rational curves.

The locus covered by these rational curves forms algebraically coisotropic subvarieties.

Provides a non-projective contractibility criterion for wall divisors.

Abstract

Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties that deform to general small deformations of the manifold. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors.