Riemann-Roch and smoothings of singularities
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 result. Some new formulae are obtained. The proof of the result is based on Wahl's globalization of \(X_ t\) hypothesis. In the appendix the author shows that this hypothesis is always satisfied.
isolated singularities of complex spaces
scientific article; zbMATH DE number 4067329