Heat Kernel on Analytic Subvariety

In this paper, the author extends Peter Li and Tian Gang''s resultson the heat kernel from projective varieties to analytic varieties.The author gets an upper bound of the heat kernel on analyticvarieties and proves several properties. Moreover, the results areextended to vector bundles. The author also gets an upper bound ofthe heat operators of some Schr"ondinger type operators on vectorbundles. As a corollary, an upper bound of the trace of the heatoperators is obtained.