Riemann-Roch theorems and elliptic genus for virtually smooth schemes
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 virtual localization formula for the virtual Euler characteristic in the presence of an equivariant action of a torus. The authors finish by showing how their results can be applied in the study of \(K\)-theoretic Donaldson invariants in moduli spaces of stable sheaves.
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