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Riemann-Roch for Non-singular Varieties

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Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 2)

Summary

The Grothendieck-Riemann-Roch theorem (GRR) states that for a proper morphism f: XY of non-singular varieties,
$$\mathrm{ch}\left ( f*\alpha \right)\cdot \mathrm{td}\left ( T_{Y} \right)=f*\left ( \mathrm{ch}\left ( \alpha \cdot \mathrm{td}\right) \left ( T_{X} \right)\right).$$
for all α in the Grothendieck group of vector bundles, or of coherent sheaves, on X. When Y is a point, one recovers Hirzebruch’s formula (HRR) for the Euler characteristic of a vector bundle E on X:
$$\sum \left ( -1 \right)^{i}\mathrm{dim}H^{i}\left ( X,E \right)=\int_{x}\mathrm{ch}\left ( E \right)\cdot \mathrm{td}\left ( T_{X} \right)$$
.

Keywords

Exact Sequence Vector Bundle Line Bundle Normal Bundle Chern Class 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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