Seungjae Lee, Symmetric Differentials on Complex Hyperbolic Forms

Let Γ be a cocompact torsion-free lattice in the automorphism group of complex unit ball Bⁿ, Aut(Bⁿ). In this talk, we discuss the existence of symmetric differentials on the compact ball quotient Σ = Bⁿ / Γ. Since Σ has a Kähler metric induced by the Bergman metric on the complex unit ball Bⁿ, it has symmetric differentials on S^mT_Σ* if m is sufficiently large. Unfortunately, finding the smallest degree m which guarantees a symmetric differential on S^mT_Σ* is difficult in even compact ball quotient cases. Instead of this, I will prove that m ≥ n+2 is a sufficient condition to give a symmetric differential on S^mT_Σ*. To achieve this goal, I will explain how to induce symmetric differentials by using a recursive formula for ∂-operators and Poincaré series. This is joint work with A. Seo.