Deformations of rational curves on primitive symplectic varieties and applications

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 that 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.

Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

“…To prove that ‚ is prime exceptional we use standard techniques on deformations of maps from rational curves to hyper-Kähler manifolds, following [53, Section 5.1] or also [19,Section 3]. We include a proof because the setting of Markman is different and because the proof in [19, Section 3] is for projective families of hyper-Kähler manifolds.…”