Non-hyperbolicity of holomorphic symplectic varieties

Abstract

We prove non-hyperbolicity of primitive symplectic varieties with $b_2\ge 5$ that satisfy the rational SYZ conjecture. If in addition $b_2\ge 7$, 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.

26 pages. We reformatted the article in the style of the journal Epiga

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