Non-hyperbolicity of holomorphic symplectic varieties

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.