Differential forms and heat diffusion on one-dimensional singular varieties. (English) Zbl 0722.58003

This paper considers one-dimensional varieties. An analytic representation of the singular homology or cohomology for these varieties is obtained. In spite of the fact that Poincaré duality does not hold for usual singular homology, the author still can construct a Hodge theory and an \(L^ 2\)-cohomology which represents the singular cohomology. At first the author defines a de Rham complex, and then deduces Hodge theory from it. The next step is to define the heat kernels on forms and to prove the index theorem. An explicit form for the parametrix of the heat kernel around singular points of the variety is given and used for small time asymptotics. It is shown that the singular points behave as Dirac mass of curvature.
Reviewer: Z.H.Guo (Beijing)


58A14 Hodge theory in global analysis
58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis