A virtual Kawasaki-Riemann-Roch formula (Q2510068)

The Hirzebruch-Riemann-Roch formula \(\chi(X, \mathcal{E})=\int_X \mathrm{ch}(\mathcal{E}) \mathrm{td}(X)\) initially proved in [\textit{F. Hirzebruch}, Neue topologische Methoden in der algebraischen Geometrie. Berlin etc.: Springer-Verlag (1956; Zbl 0070.16302)] computes the Euler characteristic of a vector bundle \(\mathcal{E}\) on a smooth projective manifold \(X\) over a field of characteristic zero. This formula has been generalized to a huge range of contexts, the most famous one being the Grothendieck-Riemann-Roch theorem, see [\textit{A. Borel} and \textit{J.-P. Serre}, Bull. Soc. Math. Fr. 86, 97--136 (1958; Zbl 0091.33004)]. The HRR formula has been extended to orbifolds by \textit{T. Kawasaki} [Osaka J. Math. 16, 151--159 (1979; Zbl 0405.32010)], and more generally the GRR theorem has been extended to algebraic stacks by \textit{B. Toen} [\(K\)-Theory 18, No. 1, 33--76 (1999; Zbl 0946.14004)]. In the present paper, the author deals with algebraic orbifolds, i.e., Deligne-Mumford stacks, admitting a perfect obstruction theory as defined by \textit{K. Behrend} and \textit{B. Fantechi} [Invent. Math. 128, No. 1, 45--88 (1997; Zbl 0909.14006)]. Then Kawasaki's formula is proved to remain valid when all the objects (structure sheaves, tangent and normal bundles) are replaced by their virtual counterpart in Fantechi-Göttsche's theory.