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We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tians partial $C^0$-estimate.
In this paper, we study questions of Demailly and Matsumura on the asymptotic behavior of dimensions of cohomology groups for high tensor powers of (nef) pseudo-effective line bundles over non-necessarily projective algebraic manifolds. By generalizi
It is proved that for any domain in $mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the multidimensional Suita conjectur
It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. Th
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
In this paper, we study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed submanifold in a complex manifold, we consider