Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture

TL;DR

This paper proves finiteness results for families of primitive symplectic varieties over pointed curves, extending the geometric Shafarevich conjecture to hyper-K"ahler manifolds.

Contribution

It establishes finiteness of isomorphism classes of such families and their fibers, assuming semi-ampleness of certain divisors, with optimality demonstrated through counterexamples.

Findings

Finiteness of isomorphism classes of generic fibers in families over pointed curves.

Finiteness of projective families up to isomorphism under semi-ampleness assumptions.

Existence of infinitely many non-isomorphic families with isomorphic fibers over the base point.

Abstract

We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve $(B, 0)$ and any fixed primitive symplectic variety $X$, among all locally trivial families of $\mathbb{Q}$-factorial and terminal primitive symplectic varieties over $B$ whose fiber over $0$ is isomorphic to $X$, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-K\"ahler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-K\"ahler varieties over some pointed curve $(B, 0)$ such that they are all isomorphic over the punctured curve $B\backslash \{0\}$ and have isomorphic fibers over the base point $0$.