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.

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

• 14B07 - Deformations of singularities
• 14J10 - Families, moduli, classification: algebraic theory
• 32Q45 - Hyperbolic and Kobayashi hyperbolic manifolds
• 32S45 - Modifications; resolution of singularities (complex-analytic aspects)
• 53C26 - Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry