Non-hyperbolicity of holomorphic symplectic varieties

We prove non-hyperbolicity of primitive symplectic varieties with b₂ ≥ 5 that satisfy the rational SYZ conjecture. If in addition b₂ ≥ 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