Jihun Yum, Limits of Bergman kernels on a Tower of Coverings of Compact Kähler Manifolds

The Bergman kernel B_X, which is by the definition the reproducing kernel of the space of L² holomorphic n-forms on a n-dimensional complex manifold X, is one of the important objects in complex geometry. In this talk, we observe the asymptotics of the Bergman kernels, as well as the Bergman metric, on a tower of coverings. More precisely, we show that, for a tower of finite Galois coverings {ϕ_j : X_j → X} of compact Kähler manifold X converging to an infinite Galois covering ϕ : X^~ → X, the sequence of push-forward Bergman kernels ϕ_j*B_Xj locally uniformly converges to ϕ_*B_X^~. Also, we show that if the canonical line bundle K_X^~ of X^~ is very ample, then the canonical line bundle K_Xj of X_j is also very ample for sufficiently large j. This is a joint work with S. Yoo in IBS-CCG.