Bergman Kernels

Abstract

Applications of the L2 method to the Bergman kernels will be discussed. Emphasis is put on the results obtained in recent decades. Among them, there are various estimates for the Bergman kernel from below on weakly pseudoconvex domains, including the solution of a long-standing conjecture of Suita by Błocki (Invent Math 193:149–158, 2013) and Guan and Zhou (Ann Math 181:1139–1208, 2015). Recently discovered variational properties due to Maitani and Yamaguchi (Math Ann 330:477–489, 2004) and Berndtsson (Ann Inst Fourier (Grenoble) 56(6):1633–1662, 2006; Ann Math 169:531–560, 2009) are also discussed. In a broader framework, they are describing the parameter dependence of the Bergman kernels associated to families or sequences of complex manifolds and vector bundles. Most of these new results are closely related to the L2 extension theorems in the previous chapter. Among them, a surprise is that a variational property of the relative canonical bundles generalizing that of the Bergman kernels, which originally belongs to the theory of variation of Hodge structures, happens to imply an optimal L2 extension theorem (cf. Berndtsson and Lempert, J Math Soc Jpn 68:1461–1472, 2016.