Gillet, Henri; Soulé, Christophe An arithmetic Riemann-Roch theorem. (English) Zbl 0777.14008 Invent. Math. 110, No. 3, 473-543 (1992). This paper contains full details of the fundamental results announced in two previous papers [see the authors, C. R. Acad. Sci., Paris, Sér. I 309, No. 17, 929-932 (1989; Zbl 0732.14002) and H. Gillet in Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 403-412 (1991; see the preceding review)]. Given an arithmetic variety \(X\) together with a Hermitian metric on its set of complex points, the “absolute” arithmetic Riemann-Roch theorem computes the degree (in the sense of Arakelov) of the determinant of the cohomology of a Hermitian vector bundle on \(X\) equipped with the Quillen metric. The computation is in terms of characteristic classes, which are invariants of the Hermitian bundle and \(X\), in the arithmetic Chow ring of \(X\). Actually, as in the case of the usual Grothendieck-Riemann-Roch theorem in algebraic geometry, the authors state and prove the “relative” version of this theorem. They use some ideas of Arakelov, Faltings and Deligne when they develop all these fundamental arithmetic concepts in arbitrary dimension. Reviewer: L.Bădescu (Bucureşti) Cited in 13 ReviewsCited in 99 Documents MSC: 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C40 Riemann-Roch theorems Keywords:arithmetic variety; Hermitian metric; arithmetic Riemann-Roch theorem; arithmetic Chow ring Citations:Zbl 0777.14007; Zbl 0732.14002 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] [BFM] Baum, P., Fulton, W., MacPherson R.: Riemann-Roch for singular varieties. Publ. Math., Inst. Hautes Etud. Sci.45, 101 145 (1975) · Zbl 0332.14003 · doi:10.1007/BF02684299 [2] [BGI] Grothendieck, A., Berthelot, P., Illusie, L.: SGA6, Théorie des intersections et théorème de Riemann-Roch. (Lect. Notes Math., vol 225) Berlin Heidelberg New York: Springer 1971 [3] [B] Bismut, J.-M.: Superconnection currents and complex immersions. Invent. Math.99, 59-113 (1990) · Zbl 0696.58006 · doi:10.1007/BF01234412 [4] [BGS1] Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles I, II, III. Commun. Math. Phys.115, 49-78, 79-126, 301-351 (1988) · Zbl 0651.32017 · doi:10.1007/BF01238853 [5] [BGS2] Bismut, J.-M., Gillet, H. et Soulé, C.: Bott-Chern currents and complex immersions. Duke Math. J.60, 255-284 (1990) · Zbl 0697.58005 · doi:10.1215/S0012-7094-90-06009-0 [6] [BGS3] Bismut, J.-M., Gillet, H., Soulé, C.: Complex immersions and Arakelov geometry. In. Grothendieck Festschrift I, pp. 249-331. Boston Basel Stuttgart: Birkhäuser, 1990 · Zbl 0744.14015 [7] [BL] Bismut J.-M., Lebeau, G.: Complex immersions and Quillen metrics. Publ. Math., Inst. Hautes Étud. Sci. (to appear) (see also ?Immersions complexes et métriques de Quillen?. C.R. Acad. Sci, Paris309, 487-491 (1989)) · Zbl 0681.53034 [8] [BV] Bismut, J.-M., Vasserot, E.: The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle. Commun. Math. Physics125, 355-367 (1989) · Zbl 0687.32023 · doi:10.1007/BF01217912 [9] [Bo] Bombieri, E.: The Mordell conjecture revisited. Ann. Scuola Mormale Superione Pisa Serie IV,17, 2, 615-640 (1990) · Zbl 0722.14010 [10] [BC] Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math.114, 71-112 (1968) · Zbl 0148.31906 · doi:10.1007/BF02391818 [11] [D] Deligne, P.: Le déterminant de la cohomologie. In: Ribet, K.A. (ed.) Current Trends in Arithmetical Algebraic Geometry. (Contemp. Math. vol. 67, pp. 93-178) Providence, RI: Am. Math. Soc. 1987 [12] [EGA1] Dieudonné, J. Grothendieck, A.: Éléments de Géométrie Algébrique I (Grund. Math. Wiss., vol. 166) Berlin Heidelberg New York: Springer 1971 [13] [EGA2] Dieudonné, J., Grothendieck, A.: Eléments de Géométrie Algébrique II. Publ. Math., Inst. Hautes Étud. Sci.8 (1961) [14] [EGA4] Dieudonné, J., Grothendieck, A.: Éléments de Géométrie Algébrique IV. Publ. Math. Inst. Hautes Étud. Sci.20 (1964) [15] [Do] Donaldson, S.K.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 3,50, 1-26 (1985) · doi:10.1112/plms/s3-50.1.1 [16] [E] Elkik, R.: Métriques sur les fibrés d’intersection. Duke Math. J.61, 303-328 (1990) · Zbl 0706.14008 · doi:10.1215/S0012-7094-90-06113-7 [17] [F1] Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387-424 (1984) · Zbl 0559.14005 · doi:10.2307/2007043 [18] [F2] Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math. 2133, 549-576 (1991) · Zbl 0734.14007 · doi:10.2307/2944319 [19] [F3] Faltings, G.: Lectures on the arithmetic Riemann Roch theorem, notes by S. Zhang. Ann. Math. Studies 127 (1992) [20] [Fr] Franke, J.: Riemann Roch in functorial form. (Preprint 1990) [21] [Fu1] Fulton, W.: Rational equivalence on singular varieties. Publ. Math. Inst. Hautes Étud. Sci.45, 147-167 (1975) · Zbl 0332.14002 · doi:10.1007/BF02684300 [22] [Fu2] Fulton, W.: Intersection Theory. (Ergeb. Math. Grenzgeb., 3. Folge, Band 2) Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005 [23] [G1] Gillet, H.: Riemann-Roch for higher algebraic K-theory. Adv. Math.40, 203 289 (1981) · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0 [24] [G2] Gillet, H.: Homological descent for the K-theory of coherent sheaves. In: Bak, A. (ed.) Algebraic K-Theory, Number Theory, Geometry and Analysis) (Lect. Notes Math., vol. 1046, pp. 80-104) Berlin Heidelberg New York: Springer 1984 [25] [G3] Gillet, H.: A Riemann-Roch theorem in arithmetic geometry. In. Proceedings International Congress of Mathematicians, Kyoto, 1990, 403 413, Springer [26] [GS1] Gillet, H., Soulé, C.: Intersection theory using Adams operations. Inven Math.90, 243-278 (1987) · Zbl 0632.14009 · doi:10.1007/BF01388705 [27] [GS2] Gillet, H. and Soulé, C.: Arithmetic intersection theory. Publ. Math., Inst. Hautes Étud. Sci.72, 94-174 (1990) · Zbl 0741.14012 · doi:10.1007/BF02699132 [28] [GS3] Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with Hermitian metrics, I, II. Ann. Math.131, 163-203 and 205 238 (1990) · Zbl 0715.14018 · doi:10.2307/1971512 [29] [GS4] Gillet, H., Soulé, C.: Analytic torsion and the arithmetic Todd genus, with an Appendix by D. Zagier. Topology30, n1, 21-54 (1991) · Zbl 0787.14005 · doi:10.1016/0040-9383(91)90032-Y [30] [GS5] Gillet, H., Soulé, C.: On the number of lattice points in convex symmetric bodies and their duals, Isr. J. Math.74, 347 357 (1991) · Zbl 0752.52008 · doi:10.1007/BF02775796 [31] [GS6] Gillet, H. and Soulé, C.: Amplitude arithmétique. Note C.R. Acad. Sci. Paris. Sér. I307, 887-890 (1988) · Zbl 0676.14007 [32] [GS7] Gillet, H., Soulé, C.: Un théorème de Riemann Roch Grothendieck arithmétique, Note CRAS ParisI, 309, 929 932 (1989) · Zbl 0732.14002 [33] [GH] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Chichester: John Wiley and Sons 1978 · Zbl 0408.14001 [34] [H1] Hartshorne, R.: Ample vector bundles. Publ. Math. Inst. Hautes Étud. Sci.29, 63 94 (1966) · Zbl 0173.49003 [35] [H2] Hartshorne, R.: Algebraic Geometry. (Grad. Texts Math., vol. 52) Berlin Heidelberg New York: Springer 1977 [36] [Hz] Hirzebruch, F.: Topological methods in algebraic geometry. (Grundl. Math., vol. 131) Berlin Heidelberg New York: Springer 1966 · Zbl 0138.42001 [37] [KM] Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves I: Preliminaries on ?det? and ?div?. Math. Scand.39, 19 55 (1976) · Zbl 0343.14008 [38] [L] Lafforgue, L.: Une version en géométrie diophantienne du ?Lemme de l’indice?. Eco. Norm. Supér. (Preprint 1990) [39] [Q1] Quillen, D.: Higher Algebraic K-Theory I. (Lect. Notes Math., vol. 341, pp. 85-147) Berlin Heidelberg New York: 1973 · Zbl 0292.18004 · doi:10.1007/BFb0067053 [40] [Q2] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31-34 (1985) · Zbl 0603.32016 · doi:10.1007/BF01086022 [41] [Q3] Quillen, D.: Superconnections and the Chern character. Topology24, 89 95 (1985) · Zbl 0569.58030 [42] [RS] Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math.98, 154-177 (1973) · Zbl 0267.32014 · doi:10.2307/1970909 [43] [S1] Soulé, C.: Opérations en K-théorie algébrique. Can. J. Math. 37, no. 3 (1985), 488-550 · Zbl 0575.14015 · doi:10.4153/CJM-1985-029-x [44] [S2] Soulé, C.: Géométrie d’Arakelov des surfaces arithmétiques. In: Séminaire, Bourbaki 713. Astérisque 177-178, 327-343 (1989) [45] [S3] Soulé, C.: Geométrie d’Arakelov et théorie des nombres transcendants, Astérisque 198-199-200, 355-373 (1991) [46] [V] Vardi, I.: Determinant of Laplacians and multiple gamma functions. SIAM J. Math. Anal.19, 493-507 (1988) · Zbl 0641.33003 · doi:10.1137/0519035 [47] [Vo] Vojta, P.: Siegel’s Theorem in the compact case. Ann. Math.133, 509-548 (1991) · Zbl 0774.14019 · doi:10.2307/2944318 [48] [Z] Zhang, S.: Ample Hermitian line bundles on arithmetic surfaces. (Preprint 1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. 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