Non-hyperbolicity of holomorphic symplectic varieties

2022/12/21 by Ljudmila Kamenova, Christian Lehn · 4 citations
Mathematics · #math.AG #math.CV #math.DG

paper · pdf · doi:10.46298/epiga.2025.11015

Abstract

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.

Cited by

• Lagrangian fibrations on Nikulin-type orbifolds
• More counterexamples to the Arithmetic Puncturing Problem
• On anti-hyperbolicity for hyperkähler varieties
• The SYZ conjecture for singular moduli spaces of sheaves on K3 surfaces

Related

• Footnotes to the birational geometry of primitive symplectic varieties
• Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces
• Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation
• Symplectically hyperbolic manifolds
• Weakly Special Manifolds with no rational curves