Estimates for the Bergman Kernel and the Multidimensional Suita Conjecture

Abstract: We study the lower bound for the Bergman kernel in terms of volume of sublevel sets of the pluricomplex Green function. We show that it implies a bound in terms of volume of the Azukawa indicatrix which can be treated as a multidimensional version of the Suita conjecture. We also prove that the corresponding upper bound holds for convex domains and discuss it in bigger detail on some convex complex ellipsoids.