Deformations of rational curves on primitive symplectic varieties and applications

TL;DR

This paper investigates how rational curves deform on primitive symplectic varieties, extending known results to singular cases and applying these to the existence and invariance of certain divisors.

Contribution

It extends deformation invariance results for prime exceptional divisors to singular primitive symplectic varieties and provides new existence results for uniruled ample divisors.

Findings

Rational curves deform along their Hodge locus in singular primitive symplectic varieties.

Prime exceptional divisors are deformation invariant along their Hodge locus in this setting.

Existence of uniruled ample divisors on primitive symplectic varieties as deformations of moduli spaces of semistable objects.

Abstract

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.