Riemann-Roch and smoothings of singularities
type: Item
instance of: scholarly article
title: Riemann-Roch and smoothings of singularities
zbMATH Open document ID: 0654.32007
DOI: 10.1016/0040-9383(86)90045-5
author: Eduard Looijenga
published in: Topology
publication date: 1986-01-01T00:00:00Z
review text: Let \(\pi\) : \(\tilde X\to X\) be a resolution of an isolated singularity x of a contractible Stein space X of dimension n. The author shows that \(\tilde X\) as well as any smoothing \(X_ t\) of (X,x) admits Chern numbers. Since neither \(\tilde X\) nor \(X_ t\) is compact, this is not a trivial result. Its proof is based on a result of O. Gabber which implies that \(H\quad *(\tilde X,{\mathfrak Q})\to H\quad *(\tilde X-\pi^{- 1}(x),{\mathfrak Q})\) is the zero map in dimension \(\geq n.\) Let f: \({\mathcal H}\to \Delta\) be a (good) representative of a smoothing \(X_ t\) of (X,x) and let \({\mathcal F}\) denote one of the natural sheaves \({\mathcal O}\), \(\Omega\), \(\Theta\) or its relative analogues. The canonical homomorphism \[ \alpha: {\mathcal O}_ X\otimes {\mathcal F}_{{\mathcal H}/\Delta}| X-\{x\}\quad \to \quad \pi_*{\mathcal F}_{\tilde X}| X-x \] extends to \({\mathcal O}_ X\)-homomorphisms \(\alpha_{{\mathcal F}}: {\mathcal J}\otimes {\mathcal F}_{{\mathcal H}/\Delta}\to \pi_*{\mathcal F}_{\tilde X}\) and \(\beta_{{\mathcal F}}: {\mathcal J}\otimes {\mathcal F}_{\tilde X/\Delta}\to {\mathcal O}_ X\otimes {\mathcal F}_{{\mathcal H}/\Delta}\quad\) for some coherent ideal sheaf \({\mathcal J}\) supported in x. Both of these homomorphisms have finite kernel and cokernel; hence their indices \(ind(\alpha_{{\mathcal F}})\) and \(ind(\beta_{{\mathcal F}})\) are defined. The difference ind(\({\mathcal F})=ind(\alpha_{{\mathcal F}})-ind(\beta_{{\mathcal F}})\) is independent of \({\mathcal J}\). The main result of the paper gives a formula for it: ind(\({\mathcal F})=P_{{\mathcal F}}[X_ t]-P_{{\mathcal F}}[\tilde X]+\sum_{i\geq 1}(-1)\) ih i (\({\mathcal F}_{\tilde X})\) where \(P_{{\mathcal F}}[Z]=\chi ({\mathcal F}_ Z)\) for some degree 2n polynomial in Chern classes. As applications and examples the author shows that many previously known formulae relating the numerical invariants of (X,x) and the invariants of its smoothing \(X_ t\) follow from his resul…
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R45
zbMATH DE Number: 4067329
zbMATH Keywords: isolated singularities of complex spaces
zbMATH Keywords: Chern number
zbMATH Keywords: Riemann-Roch defect theorem
zbMATH Keywords: smoothing
MaRDI profile type: Publication
full work available at URL: 0040-9383(86)90045-5
OpenAlex ID: W2071702965
description: scientific article; zbMATH DE number 4067329