A stochastic approach to the Riemann-Roch theorem
type: Item
instance of: scholarly article
title: A stochastic approach to the Riemann-Roch theorem
zbMATH Open document ID: 0709.58036
author: Ichiro Shigekawa
author: Naomasa Ueki
published in: Osaka Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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: \(d=\partial +{\bar\partial}\).
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4166123
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: index of Dolbeault complex
zbMATH Keywords: holomorphic vector bundle
zbMATH Keywords: compact Kähler manifold
zbMATH Keywords: spaces of complex differential forms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4166123