The second coefficient of the asymptotic expansion of the weighted   Bergman kernel for $(0,q)$ forms on $\Complex^n$
  Let $\phi\in C^\infty(\Complex^n)$ be a given real valued function. We assume that $\pr\ddbar\phi$ is non-degenerate of constant signature $(n_-,n_+)$ on $\Complex^n$. 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. 
Chin-Yu Hsiao
math.CV math.DG