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A singular Riemann-Roch for Hirzebruch characteristics. (English) Zbl 0915.14005

Jakubczyk, Bronisław (ed.) et al., Singularities symposium – Łojasiewicz 70. Papers presented at the symposium on singularities on the occasion of the 70th birthday of Stanisław Łojasiewicz, Cracow, Poland, September 25–29, 1996 and the seminar on singularities and geometry, Warsaw, Poland, September 30–October 4, 1996. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 44, 257-268 (1998).
The Hirzebruch-Riemann-Roch theorem (HRR) [F. Hirzebruch, “Topological methods in algebraic geometry” (1966; Zbl 0138.42001)] for vector bundles on a non-singular complex variety was generalized by Grothendieck (GRR) [see A. Borel and J.-P. Serre, Bull. Soc. Math. Fr. 86, 97-136 (1959; Zbl 0091.33004)] and further extended to singular varieties by Baum, Fulton and MacPherson (BFM-RR) [see P. Baum, W. Fulton and R. MacPherson, Acta Math. 143, 155-192 (1979; Zbl 0474.14004)]. The HRR says, symbolically speaking, that \(\chi=T\), where \(\chi\) denotes the Euler-Poincaré characteristic of the bundle and \(T\) denotes its Todd characteristic. Hirzebruch generalized these two characteristics to \(\chi_y\) and \(T_y\), introducing a parameter \(y\) [see the cited book by F. Hirzebruch; §21.3] and he showed that \(\chi_y=T_y\). In this paper we give a “BFM-RR version” of this generalized HRR.
For the entire collection see [Zbl 0906.00013].

MSC:

14C40 Riemann-Roch theorems
14B05 Singularities in algebraic geometry
14F99 (Co)homology theory in algebraic geometry
18F99 Categories in geometry and topology
57R20 Characteristic classes and numbers in differential topology
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