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The Riemann-Roch Theorem

In: A Royal Road to Algebraic Geometry

Author

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  • Audun Holme

    (University of Bergen, Department of Mathematics)

Abstract

This chapter is on the Riemann-Roch Theorem. We start with Hirzebruch’s Riemann-Roch Theorem, and deduce from it the Riemann-Roch Theorem for curves, and for surfaces. We state the general Grothendieck’s Riemann-Roch theorem, deducing that of Hirzebruch from it. Some general constructions and concepts used in this chapter have been extended so that schemes with singularities are covered. This theory is developed in the paper (Publ. Math. l’HIES, 45:147–167, 1975) and in the book (Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 1st edn., vol. 2. Springer, Berlin, 1984 and 2nd edn. 1998) by W. Fulton. One of the most important applications of this marvelous work is the general Baum-Fulton-McPehrson Riemann-Roch Theorem for singular varieties (in Baum et al., Publ. Math. l’HIES, 45, 101–145, 1975). However, we do not include a survey of this work here, as it reaches beyond the scope of the present book. The non singular case is challenging enough at this stage, but the theorem proved in Baum et al. (Publ. Math. l’HIES, 45, 101–145, 1975) would be an exiting source for further study!

Suggested Citation

  • Audun Holme, 2012. "The Riemann-Roch Theorem," Springer Books, in: A Royal Road to Algebraic Geometry, chapter 0, pages 329-334, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-19225-8_20
    DOI: 10.1007/978-3-642-19225-8_20
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