The log term of the Szegö kernel
The setting for this article is a positive Hermitian line bundle~\((L,h)\) over a compact complex manifold~\(M\) of dimension~\(n\). The authors study the asymptotic expansion of the Szegő kernel function for the unit circle bundle of the dual bundle~\(L^*\). They show that for a generic metric~\(h\), the logarithmic term of the Szegő kernel is not zero. When \(M\)~is complex projective space \(\mathbb{C}P^n\) and \(L\)~is the hyperplane bundle, they prove a local rigidity theorem: namely, if \(h\)~is the standard metric whose curvature is the Fubini-Study metric~\(\omega\), then for any metric~\(h'\) sufficiently close to~\(h\) for which the logarithmic term of the Szegő kernel of the corresponding unit circle bundle is zero, there is an automorphism~\(f\) of \(\mathbb{C}P^n\) such that the curvature of~\(h'\) pulled back by~\(f\) equals~\(\omega\). Moreover, the authors prove a corresponding global rigidity theorem when \(n=1\). They conjecture that the global rigidity holds for arbitrary~\(n\); this is an analogue for the Szegő kernel of the Ramadanov conjecture for the Bergman kernel.
scientific article; zbMATH DE number 2139675
scientific article