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A virtual Kawasaki-Riemann-Roch formula. (English) Zbl 1346.19009

The Hirzebruch-Riemann-Roch formula \(\chi(X, \mathcal{E})=\int_X \mathrm{ch}(\mathcal{E}) \mathrm{td}(X)\) initially proved in [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 [A. Borel and J.-P. Serre, Bull. Soc. Math. Fr. 86, 97–136 (1958; Zbl 0091.33004)]. The HRR formula has been extended to orbifolds by 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 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 K. Behrend and 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.

MSC:

19L10 Riemann-Roch theorems, Chern characters
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