A stochastic approach to the Riemann-Roch theorem
In this paper, following Watanabe's method a probabilistic proof of the Riemann-Roch theorem is given, i.e. the computation of the index of Dolbeault complex with coefficients in a holomorphic vector bundle V: \[ {\bar \partial}_ v: \Gamma^{\infty}(\Lambda^{0,+}(M)\otimes V)\to \Gamma^{\infty}(\Lambda^{0,-}(M)\otimes V) \] where M is a compact Kähler manifold \(\Lambda^{0,+}(M)\) and \(\Lambda^{0,-}(M)\) are spaces of complex differential forms of the degree (0,even) and (0,odd), respectively, V is a holomorphic vector bundle and \({\bar \partial}\) is the usual \({\bar \partial}\)-operator obtained by a decomposition of the exterior differential d: \(d=\partial +{\bar\partial}\).
scientific article; zbMATH DE number 4166123