Tag: Seungjae Lee

As the continuation of the previous talk, I discuss an L² extension problem of holomorphic jets on compact complex hyperbolic forms. Let Γ be a cocompact torsion-free lattice in the automorphism group Aut(Bⁿ) and Ω be a quotient Bⁿ × Bⁿ given by diagonal action of Γ. In the setting, Ω becomes a ball-fiber bundle over Σ = Bⁿ / Γ. Since we can identify symmetric differentials on Σ and jets of holomorphic function on D which is the maximal compact analytic variety on Ω, it is natural to expect that holomorphic function on Ω can be derived by symmetric differentials. In this context, M. Adachi (2017) extends holomorphic jets on D to weighted L² holomorphic functions on Σ for the n=1 case. In 2020, A. Seo and S. Lee generalized his result by developing a Hodge type identity on S^mT_Σ*. In this talk, I will explain recent progress and if time is permitted, I sketch the proof of our result. This is joint work with A. Seo.

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.