Direct images in non-archimedean Arakelov theory
Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 363-399.

Nous développons un formalisme d’images directes pour les fibrés hermitiens dans le contexte de la théorie d’Arakelov non-archimédienne que nous avons introduite avec S. Bloch. Nous montrons un théorème de Riemann-Roch-Grothendieck pour cette image directe.

We develop a formalism of direct images for metrized vector bundles in the context of the non-archimedean Arakelov theory introduced in our joint work with S. Bloch. We prove a Riemann-Roch-Grothendieck theorem for this direct image.

@article{AIF_2000__50_2_363_0,
     author = {Gillet, Henri and Soul\'e, Christophe},
     title = {Direct images in non-archimedean {Arakelov} theory},
     journal = {Annales de l'Institut Fourier},
     pages = {363--399},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     doi = {10.5802/aif.1758},
     zbl = {0969.14015},
     mrnumber = {2001j:14036},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1758/}
}
TY  - JOUR
AU  - Gillet, Henri
AU  - Soulé, Christophe
TI  - Direct images in non-archimedean Arakelov theory
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 363
EP  - 399
VL  - 50
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1758/
DO  - 10.5802/aif.1758
LA  - en
ID  - AIF_2000__50_2_363_0
ER  - 
%0 Journal Article
%A Gillet, Henri
%A Soulé, Christophe
%T Direct images in non-archimedean Arakelov theory
%J Annales de l'Institut Fourier
%D 2000
%P 363-399
%V 50
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1758/
%R 10.5802/aif.1758
%G en
%F AIF_2000__50_2_363_0
Gillet, Henri; Soulé, Christophe. Direct images in non-archimedean Arakelov theory. Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 363-399. doi : 10.5802/aif.1758. https://aif.centre-mersenne.org/articles/10.5802/aif.1758/

[AKMW] D. Abramovich, K. Karu, K. Matsuki, J. Wlodarczyk, Torification and factorization of birational maps, preprint, 1999, math.AG/9904135. | Zbl

[BFM] P. Baum, W. Fulton, R. Macpherson, Riemann-Roch for Singular Varieties, Pub. Math. I.H.E.S., 45 (1975), 253-290. | Numdam | MR | Zbl

[BK] J.-M. Bismut, K. Koehler, Higher analytic torsion forms for direct images and anomaly formulas, J. Algebr. Geom., 1, n° 4 (1992), 647-684. | MR | Zbl

[BGS] S. Bloch, H. Gillet, C. Soulé, Non-archimedean Arakelov theory, Journal of Algebraic Geometry, 4 (1995), 427-485. | MR | Zbl

[B] J. Burgos, Arithmetic Chow rings and Deligne-Beilinson cohomology, J. Algebr. Geom., 6, n° 2 (1997), 335-377. | MR | Zbl

[D] P. Deligne, Le déterminant de la cohomologie, in : Current trends in Arithmetical Algebraic Geometry, K. A. Ribet ed., Contemporary Math., 67 (1987), 93-178. | MR | Zbl

[F] W. Fulton, Intersection theory, Ergebnisse der Math., 3, Folge 2 Band 2, Springer-Verlag, Berlin-Heidelberg-New York, 1984. | MR | Zbl

[Fr] J. Franke, Riemann-Roch in functorial form, preprint, 78 pp., 1992.

[GS1] H. Gillet, C. Soulé, Arithmetic Intersection Theory, Publications Math. IHES, 72 (1990), 94-174. | Numdam | MR | Zbl

[GS2] H. Gillet, C. Soulé, Characteristic classes for algebraic vector bundles with hermitian metric, Annals of Math., 131 (1990), 163-203. | MR | Zbl

[GS3] H. Gillet, C. Soulé, Analytic torsion and the Arithmetic Todd genus, Topology, 30, 1 (1991), 21-54. | MR | Zbl

[GS4] H. Gillet, C. Soulé, An arithmetic Riemann-Roch theorem, Inventiones Math., 110 (1992), 474-543. | MR | Zbl

[H] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math., 79 (1964), 109-326. | MR | Zbl

[KM] F. F. Knudsen, D. Mumford, The projectivity of the moduli space of stable curves, I: Preliminaries on “det” and “div”, Math. Scand., 39 (1976), 19-55. | MR | Zbl

[M] H. Matsumura, Commutative ring theory, Transl. from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1989. | Zbl

[Q] D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl., (1985), 31-34. | MR | Zbl

[RG] M. Raynaud, L. Gruson, Critères de platitude et de projectivité, Inv. Math., 13 (1971), 1-89. | Zbl

[S] T. Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. Journal, 57 (1988), 151-173. | MR | Zbl

[SGA4] M. Artin, A. Grothendieck, J.-L. Verdier, P. Deligne, B. Saint-Donat, Séminaire de géométrie algébrique du Bois-Marie 1963-1964, Théorie des topos et cohomologie étale des schémas, SGA 4, Tome 3, Exposés IX a XIX, Lecture Notes in Mathematics, Berlin-Heidelberg-New York, Springer-Verlag, 305 (1973). | MR | Zbl

[SGA6] P. Berthelot, A. Grothendieck, L. Illusie, Séminaire de géométrie algébrique du Bois Marie 1966/67, SGA 6, Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Berlin-Heidelberg-New York, Springer-Verlag, 225 (1971). | Zbl

[W] J. Wlodarczyk, Combinatorial structures on toroidal varieties and a proof of the weak factorization theorem preprint, 1999, math.AG/9904076.

[Z] J. Zha, A general Arithmetic Riemann-Roch theorem, PHD thesis, Chicago University, 1998.

Cité par Sources :