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The Second Coefficient of the Asymptotic Expansion of the Weighted Bergman Kernel for (0,q) Forms on $\mathbb{C^{n}}$
by Chin-Yu Hsiao

Vol. 11 No. 3 (2016) P.521~P.570
 DOI: https://doi.org/10.21915/BIMAS.2016302 10.21915/BIMAS.2016302

ABSTRACT

Let $\phi\in C^\infty(\mathbb{C}^n)$ be a given real valued function. We assume that $\partial\bar\partial\phi$ is non-degenerate of constant signature $(n_-,n_+)$ on $\mathbb{C}^n$. When $q=n_-$, it is well-known that the Bergman kernel for $(0,q)$ forms with respect to the $k$-th weight $e^{-2k\phi}$, $k>0$, admits a full asymptotic expansion in $k$. In this paper, we compute the trace of the second coefficient of the asymptotic expansion on the diagonal.

KEYWORDS
Bergman kernel asymptotics, Kodaira Laplacian, the method of stationary phase.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 58J40, 32C15.

MILESTONES