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arXiv · 2212.11411

Non-hyperbolicity of holomorphic symplectic varieties

Ljudmila Kamenova·Christian Lehn
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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.

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Ljudmila Kamenova, Christian Lehn. 2022-12-21. Non-hyperbolicity of holomorphic symplectic varieties. https://doi.org/10.46298/epiga.2025.11015

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