Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers …. These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties.
We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.
@article{CML_2009__1_1_1_0,
author = {Andr\'e, Yves},
title = {Slope filtrations},
journal = {Confluentes Mathematici},
pages = {1--85},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {1},
number = {1},
year = {2009},
doi = {10.1142/S179374420900002X},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S179374420900002X/}
}
André, Yves. Slope filtrations. Confluentes Mathematici, Tome 1 (2009) no. 1, pp. 1-85. doi : 10.1142/S179374420900002X. http://www.numdam.org/articles/10.1142/S179374420900002X/
[1] A. Abbes and T. Saito, Amer. J. Math. 124, 879 (2002), DOI: 10.1353/ajm.2002.0026 .
[2] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields, II, Doc. Math., (Extra Vol.): 5-72 (electronic), 2003. Kazuya Katos fiftieth birthday .
[3] C. Adams, Ann. Math. 30, 195 (1929), DOI: 10.2307/1968274 .
[4] Y. André, Invent. Math. 148, 285 (2002).
[5] Y. André, Complex Analysis, Dynamical Systems, Summability of Divergent Series and Galois Theories. I. Volume in Honor of Jean-Pierre Ramis, S.M.F. Astérisque 296, ed. M. Loday-Richaud (2004) pp. 43-53.
[6] Y. André, Astérisque 299, 115 (2005).
[7] Y. André, Invent. Math. 170, 147 (2007), DOI: 10.1007/s00222-007-0060-3 .
[8] Y. André, Sur le produit tensoriel de réseaux hermitiens semistables, in preparation .
[9] Y. André and L. Di Vizio, Complex Analysis, Dynamical Systems, Summability of Divergent Series and Galois Theories. I. Volume in Honor of Jean-Pierre Ramis, S.M.F. Astérisque 296, ed. M. Loday-Richaud (2004) pp. 55-111.
[10] Y. André and B. Kahn, Rend. Sem. Padova 108, 107 (2002).
[11] C. Arf, J. Reine Angew. Math. 181, 1 (1940).
[12] M. Atiyah and R. Bott, Phil. Trans. Roy. Soc. London 308, 523 (1983).
[13] J. Ayoub, Astérisque 314, 315 (2007).
[14] L. Barbieri-Viale, A. Rosenschon and M. Saito, Ann. Math. 158, 593 (2003), DOI: 10.4007/annals.2003.158.593 .
[15] L. Berger, Astérisque 319, 13 (2008).
[16] A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 100 (1982) 5-171 .
[17] A. Bondal and M. van der Bergh, Mosc. Math. J. 3, 1 (2003).
[18] T. Borek, J. Number Th. 113, 380 (2005), DOI: 10.1016/j.jnt.2004.12.003 .
[19] J.-B. Bost, Hermitian vector bundles and stability, talk at Oberwolfach, “Algebraische Zahlentheorie”, July 1997 .
[20] J.-B. Bost and K. Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. I. geometry of numbers, arXiv:0701343 .
[21] N. Bourbaki, Commutative Algebra, Chaps. 1-7 [translated from French, reprint of the 1989 English translation. Elements of Mathematics (Springer-Verlag, 1998)] .
[22] T. Bridgeland, Ann. Math. 166, 317 (2007), DOI: 10.4007/annals.2007.166.317 .
[23] S. Bradlow, Int. J. Math. 6, 645 (1995), DOI: 10.1142/S0129167X95000262 .
[24] L. Bruasse, Ann. I’inst Fourier 53, 541 (2003).
[25] H. Chen, Harder-Narasimhan categories , arXiv:0706.2648v2 .
[26] H. Chen, Maximal slope of tensor product of Hermitian vector bundles, preprint hal00151961v2 .
[27] F. Cherbonnier and P. Colmez, Invent. Math. 133, 581 (1998), DOI: 10.1007/s002220050255 .
[28] B. Chiarellotto and A. Pulita, Arithmetic and differential swan conductors of rank one representations with finite local monodromy , arXiv:0711.0701v2 .
[29] G. Christol and Z. Mebkhout, Ann. Math. 151, 385 (2000), DOI: 10.2307/121041 .
[30] P. Colmez, Conducteur d’Artin d’une représentaiton de De Rham, preprint IMJ, 2003 .
[31] P. Colmez and J. M. Fontaine, Inv. Math. 140, 1 (2000), DOI: 10.1007/s002220000042 .
[32] A. De Jong and F. Oort, J. A. M. S. 13, 209 (1999), DOI: 10.1090/S0894-0347-99-00322-7 .
[33] P. Deligne, Publ. Math. I.H.E.S. 40, 5 (1971).
[34] P. Deligne, Publ. Math. I.H.E.S. 44, 5 (1974).
[35] E. De Shalit and O. Parzanchevski, On tensor products of semistable lattices, preprint Jerusalem, 2006 .
[36] J. Dieudonné, Amer. J. Math. 77, 429 (1955), DOI: 10.2307/2372633 .
[37] L. Di Vizio, J. Noncomm. Geom. 3, 125 (2009).
[38] L. Di Vizio, Gazette des Mathématiciens (Société Mathématique de France, 2003) pp. 20-49.
[39] G. Faltings, J. Alg. Geom. 2, 549 (1993).
[40] G. Faltings, Mumford-stabilität in der algebraiscen geometrie, Proc. Intern. Congress Math. (Birkhäuser, 1995) pp. 648-655.
[41] G. Faltings and G. Wüstholz, Invent. Math. 116, 109 (1994), DOI: 10.1007/BF01231559 .
[42] L. Fargues, La filtration de Harder-Narasimhan des schémas en groupes finis et plats, preprint Orsay 2008 .
[43] J.-M. Fontaine, Représentations p-adiques des corps locaux I. The Grothendieck Festschrift, Vol. II, 249-309, Progr. Math. 87 (Birkhäuser, 1990) .
[44] E. Gaudron, Rend. Padova (2008).
[45] R. Gérard and A. Levelt, Ann. Inst. Fourier 157 (1973).
[46] D. Gieseker, Invent. Math. 43, 233 (1977), DOI: 10.1007/BF01390081 .
[47] A. Gorodentsev, S. Kuleshov and A. Rudakov, Stability data and t-structures on a triangulated category , arXiv:math/0312442 .
[48] D. Grayson, Higher algebraic K-theory II (after Daniel Quillen), Algebraic K-theory, Proc. Conf. Nothwestern Univ551, Springer Lect. Notes (Springer, 1976) pp. 217-240.
[49] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Univ. de Montréal 1974 .
[50] D. Happel, I. Reiten and S. Smale, Memoir A. M. S. 575, (1996).
[51] G. Harder and M. Narasimhan, Math. Ann. 215 (1974).
[52] H. Hasse and Führer, J. Reine Angew. Math. 162, 169 (1930).
[53] N. Hitchin, Proc. London Math. Soc. 55, 59 (1987).
[54] N. Hoffmann, J. Jahnel and U. Stuhler, J. Reine Angew. Math. 495, 35 (1998).
[55] M. Jurchescu, Theory of Categories (Romanian) (Editura Acad. Republicii Socialiste România, 1966), pp. 73-240 .
[56] N. Katz, Slope filtrations of F-crystals, J. Géom. Alg. Rennes, July 1978 Astérisque 63 (1979) 113-163 .
[57] N. Katz, Invent. Math. 87, 13 (1987), DOI: 10.1007/BF01389152 .
[58] N. Katz , Gauss Sums, Kloosterman Sums, and Monodromy Groups , Ann. Math. Studies 116 ( Princeton Univ. Press , 1988 ) .
[59] K. Kedlaya, Doc. Math. 10, 447 (2005).
[60] K. Kedlaya, Astérisque 319, 259 (2008).
[61] K. Kedlaya, Good formal structures on flat meromorphic connections, I: Surfaces , arXiv:0811.0190 .
[62] K. Kedlaya , p-adic Differential Equations ( 2008 ) .
[63] S. Kobayashi , Differential Geometry of Complex Vector Bundles ( Princeton Univ. Press , 1987 ) .
[64] R. Lazarsfeld, Positivity in Algebraic Geometry. II. Positivity for vector bundles, and multiplier ideals. Ergeb. der Math. und ihrer Grenzgebiete. 3. Folge. 49 (Springer-Verlag, 2004) .
[65] A. Levelt, Ark. Math. 13, 1 (1975), DOI: 10.1007/BF02386195 .
[66] R. Liu, Slope filtrations in families , arXiv:0809.0331 .
[67] S. Mac Lane , Categories for the Working Mathematician , 2nd edn. ( Springer , 1988 ) .
[68] B. Malgrange , Équations Différentielles à Coefficients Polynomiaux , Progress in Mathematics ( Birkhäuser , 1991 ) .
[69] Yu. Manin, Usp. Mat. Nauk 18, 3 (1963).
[70] A. Marmora, Doc. Math. 9, 413 (2004).
[71] M. Maruyama, Math. Ann. 255, 317 (1981), DOI: 10.1007/BF01450706 .
[72] S. Matsuda, Geometric Aspects of Dwork Theory I and II (Walter de Gruyter, 2004) pp. 845-856.
[73] V. Mehta and A. Ramanathan, Math. Ann. 258, 213 (1982), DOI: 10.1007/BF01450677 .
[74] Y. Miyaoka, The Chern Classes and Kodaira Dimension of a Minimal Variety, Algebraic Geometry, Adv. Stud. Pure Math 10 (North-Holland, Sendai, 1987) pp. 449-476.
[75] T. Mochizuki, Good formal structure for meromorphic flat connections on smooth projective surfaces , arXiv:0803.1346 .
[76] A. Moriwaki, Subsheaves of a Hermitian torsion-free coherent sheaf on an arithmetic variety , arXiv:math/0612268 .
[77] D. Mumford, Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Inst. Mittag-Leffler, Stockholm, 1963) pp. 526-530.
[78] M. Narasimhan and C. Seshadri, Ann. Math. 82, 540 (1965), DOI: 10.2307/1970710 .
[79] P. O’Sullivan, Finite dimensionality of reductive envelopes, appendix to [10] .
[80] P. O’Sullivan, C. R. Math. Acad. Sci. Paris 340, 557 (2005).
[81] C. Praagman, Proc. Kon. Ned. Ac. Wet., Ser. A 45, 249 (1983).
[82] A. Pulita, p-adic confluence of q-difference equations, to appear in Compos. Math. .
[83] M. van der Put and M. Reversat, Galois theory of q-difference equations, to appear in Ann. Fac. Sci. Toulouse .
[84] D. Rakov, Dokl. Akad. Nauk SSSR 188, 1006 (1969).
[85] S. Ramanan and A. Ramanathan, Tohoku Math. J. 36, 269 (1984), DOI: 10.2748/tmj/1178228852 .
[86] L. Ramero, Publ. Math. I.H.E.S. 102, 167 (2005), DOI: 10.1007/s10240-005-0036-z .
[87] L. Ramero, Local monodromy in non-Archimedean analytic geometry II, arXiv/0703225 .
[88] J.-P. Ramis and J. Sauloy, RIMS Kôkyûroku Bessatsu B 2, 167 (2007).
[89] M. Rapoport, Analogien zwischen den Modulräumen von Vektorbündeln und von Flaggen, DMV Tagung Ulm, 1995 .
[90] A. Rudakov, J. Alg. 197, 231 (1997), DOI: 10.1006/jabr.1997.7093 .
[91] W. Rump, Bull. London Math. Soc. (2008), DOI: 10.1112/blms/bdn080 .
[92] C. Sabbah, Astérisque 263, (2000).
[93] N. Savedra Rivano , Catégories Tannakiennes , Lecture Notes in Mathematics 265 ( Springer-Verlag , 1972 ) .
[94] J. Sauloy, Ann. L’inst. Fourier 54, 181 (2004).
[95] J. Sauloy, Equations aux q-différences et fibrés vectoriels holomorphes sur la courbe elliptique C*/qZ , arXiv:0711.4031 .
[96] J.-P. Schneiders , Quasi-Abelian Categories and Sheaves , Mém. S. M. F 76 ( Soc. Math. France , 1991 ) .
[97] J.-P. Serre , Corps Locaux ( Hermann , 1968 ) .
[98] S. Shatz, Compos. Math. 35, 163 (1977).
[99] C. Simpson, Publ. Math. IHES 75, 5 (1992).
[100] R. Succi Cruciani, Rev. Roumaine Math. Pures Appl. 18, 105 (1973).
[101] U. Stuhler, Arch. Math. 27, 604 (1976), DOI: 10.1007/BF01224726 .
[102] F. Takemoto, Nagoya Math. J. 47, 29 (1972).
[103] M. Toma, C. R. Acad. Sci. Paris 323, 501 (1996).
[104] B. Totaro, Tensor products of semistables are semistable, in Geometry and Analysis on Complex Manifolds, Festschrift for Professor S. Kobayashi’s 60th Birthday, eds. T. Noguchi, J. Noguchi and T. Ochiai (World Scientific, 1994), pp. 242-250 .
[105] B. Totaro, Duke Math. J. 83, 79 (1996), DOI: 10.1215/S0012-7094-96-08304-0 .
[106] N. Tsuzuki, Compos. Math. 111, 245 (1998).
[107] H. Turrittin, Acta Math. 93, 27 (1955), DOI: 10.1007/BF02392519 .
[108] L. Xiao, On ramification filtrations and p-adic differential equations, I: Equal characteristic case , arXiv:0801.4962 .
[109] L. Xiao, On ramification filtrations and p-adic differential equations, II: Mixed characteristic case , arXiv:0811.3792 .
[110] N. Yoneda, J. Fac. Sci. Univ. Tokyo 18, 507 (1960).
Cité par Sources :
