Non-hyperbolicity of holomorphic symplectic varieties

TL;DR

This paper proves that certain primitive symplectic varieties are non-hyperbolic and have vanishing Kobayashi pseudometric under specific conditions, extending known results to all current examples.

Contribution

It establishes non-hyperbolicity and Kobayashi pseudometric vanishing for primitive symplectic varieties satisfying the rational SYZ conjecture, including all known irreducible symplectic manifolds.

Findings

Primitive symplectic varieties with $b_2 \\geq 5$ are non-hyperbolic.

If $b_2 \\geq 7$, the Kobayashi pseudometric vanishes.

Projective primitive symplectic varieties with a Lagrangian fibration have vanishing Kobayashi pseudometric.

Abstract

We prove non-hyperbolicity of primitive symplectic varieties with $b_2 \geq 5$ that satisfy the rational SYZ conjecture. If in addition $b_2 \geq 7$, we establish that the Kobayashi pseudometric vanishes identically. This in particular applies to all currently known examples of irreducible symplectic manifolds and thereby completes the results by Kamenova--Lu--Verbitsky. The key new contribution is that a projective primitive symplectic variety with a Lagrangian fibration has vanishing Kobayashi pseudometric. The proof uses ergodicity, birational contractions, and cycle spaces.