Riemann-Roch theorems and elliptic genus for virtually smooth schemes
type: Item
instance of: scholarly article
title: Riemann-Roch theorems and elliptic genus for virtually smooth schemes
zbMATH Open document ID: 1194.14017
author: Barbara Fantechi
author: Lothar Göttsche
published in: Geometry \& Topology
publication date: 2009-11-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0706.0988
review text: A virtually smooth scheme of expected dimension \(d\) is a pair \((X,E^{\bullet})\) where \(X\) is a scheme that can be embedded in a smooth scheme and \(E^{\bullet}\) is a 1-perfect obstruction theory for \(X\) with rk \( E^{\bullet} = d\). Virtually smooth schemes were shown in \textit{J. Li} and \textit{G. Tian} [First Int. Press Lect. Ser. 1, 47--83 (1998; Zbl 0978.53136)] and \textit{K. Behrend} and \textit{B. Fantechi} [Invent. Math. 128, No. 1, 45--88 (1997; Zbl 0909.14006)] to admit a virtual fundamental class suited for doing intersection theory. In the present paper, the authors define and study virtual versions of the holomorphic Euler characteristic for elements in \(K^0(X)\) and of two refinements of this: the \(\chi_{-y}\)-genus and the elliptic genus. These are shown to be deformation invariant and to reduce to the usual definitions in the case \(X\) is smooth and \(E^{\bullet}\) is the cotangent bundle. The virtual \(\chi_{-y}\)-genus is shown to be a polynomial of degree \(d\) and, as a consequence, it is also possible to define a virtual version of the topological Euler characteristic. If \(X\) has lci singularities and \(E^{\bullet}\) is the cotangent complex \(L_X^{\bullet}\), then the topological Euler characteristic is shown to coincide with Fulton's Chern class, which is then invariant under deformations for proper lci schemes. The main results of the paper are virtual versions of the Theorems of Hirzebruch-Riemann-Roch and of Grothendieck-Riemann-Roch (in the later the target of the morphism is required to be a smooth scheme). Similar results holding for \([0,1]\)-manifolds were proven by \textit{I. Ciocan-Fontanine} and \textit{M. Kapranov} in [Geom. Topol. 13, No. 3, 1779--1804 (2009; Zbl 1159.14002)]. Using the virtual version of Grothendieck-Riemann-Roch and Graber-Pandharipande virtual localization [see \textit{T. Graber} and \textit{R. Pandharipande} [Invent. Math. 135, No. 2, 487--518 (1999; Zbl 0953.14035)], the authors establish a vi…
reviewed by: Margarida Melo
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 5624187
zbMATH Keywords: Riemann-Roch theorems
zbMATH Keywords: virtual fundamental class
zbMATH Keywords: virtual Euler characteristic
zbMATH Keywords: genus
zbMATH Keywords: localization
MaRDI profile type: Publication
OpenAlex ID: W3103051281
arXiv ID: 0706.0988
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DOI: 10.2140/GT.2010.14.83
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description: scientific article