Abstract
We sketch here a proof of the Riemann–Roch theorem.
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Notes
- 1.
In 1957 Grothendieck gave in Princeton another generalization of Riemann–Roch’s theorem that includes the result of Hirzebruch (cf. [3]). This generalisation is known as the Grothendieck–Hirzebruch–Riemann–Roch theorem.
- 2.
This label is more meaningful for the case of complex manifolds of higher dimensions. In this case, the canonical bundle is the determinant bundle of \(T^{*}X\).
- 3.
In the literature, this result, also true in higher dimensions, is called the Hodge theorem.
- 4.
- 5.
This assertion is known as the moduli problem and for more historical details we refer the reader to [1].
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Acknowledgements
The authors would like to express their sincere gratitude to Athanase Papadop-oulos for giving them the opportunity for writing this chapter and especially for his patience, kindness, and constructive remarks. This work was partially supported by the French ANR grant FINSLER (Géométrie de Finsler et applications, ANR-12-BS01-0009). The last author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
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A’Campo, N., Alberge, V., Frenkel, E. (2017). The Riemann–Roch Theorem. In: Ji, L., Papadopoulos, A., Yamada, S. (eds) From Riemann to Differential Geometry and Relativity. Springer, Cham. https://doi.org/10.1007/978-3-319-60039-0_13
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