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Good formal structures for flat meromorphic connections, II: Excellent schemes


Author: Kiran S. Kedlaya
Journal: J. Amer. Math. Soc. 24 (2011), 183-229
MSC (2010): Primary 14F10; Secondary 32C38
Published electronically: September 17, 2010
MathSciNet review: 2726603
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Abstract: Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for algebraic varieties. The argument combines a numerical criterion for good formal structures from a previous paper, with an analysis based on the geometry of an associated valuation space (Riemann-Zariski space). We obtain a similar result over the formal completion of an excellent scheme along a closed subscheme. If we replace the excellent scheme by a complex analytic variety, we obtain a similar but weaker result in which the blowup can only be constructed in a suitably small neighborhood of a prescribed point.


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  • 1. Jose M. Aroca, Heisuke Hironaka, and José L. Vicente, The theory of the maximal contact, Instituto “Jorge Juan” de Matemáticas, Consejo Superior de Investigaciones Cientificas, Madrid, 1975. Memorias de Matemática del Instituto “Jorge Juan”, No. 29. [Mathematical Memoirs of the “Jorge Juan” Institute, No. 29]. MR 0444999
  • 2. José M. Aroca, Heisuke Hironaka, and José L. Vicente, Desingularization theorems, Memorias de Matemática del Instituto “Jorge Juan” [Mathematical Memoirs of the Jorge Juan Institute], vol. 30, Consejo Superior de Investigaciones Científicas, Madrid, 1977. MR 480502
  • 3. F. Baldassarri, Continuity of the radius of convergence of differential equations on $ p$-adic analytic curves, arXiv preprint 0809.2479v6 (2010).
  • 4. F. Baldassarri and L. Di Vizio, Continuity of the radius of convergence of $ p$-adic differential equations on Berkovich analytic spaces, arXiv preprint 0709.2008v3 (2008).
  • 5. Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709
  • 6. Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306, 10.1007/s002220050141
  • 7. E. Bierstone, P. Milman, and M. Temkin, $ \mathbb{Q}$-universal desingularization, arXiv preprint 0905.3580v1 (2009).
  • 8. S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961
  • 9. Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449–494. MR 2426355, 10.2977/prims/1210167334
  • 10. Gilles Christol, Modules différentiels et équations différentielles 𝑝-adiques, Queen’s Papers in Pure and Applied Mathematics, vol. 66, Queen’s University, Kingston, ON, 1983 (French). MR 772749
  • 11. A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
  • 12. Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • 13. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
  • 14. William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
  • 15. Hans Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Études Sci. Publ. Math. 5 (1960), 64 (German). MR 0121814
  • 16. A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. MR 0217084
  • 17. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 0217085
  • 18. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181
  • 19. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 0217086
  • 20. Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446
  • 21. Heisuke Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), 503–547. MR 0393556
  • 22. Kiran S. Kedlaya, Semistable reduction for overconvergent 𝐹-isocrystals. I. Unipotence and logarithmic extensions, Compos. Math. 143 (2007), no. 5, 1164–1212. MR 2360314, 10.1112/S0010437X07002886
  • 23. Kiran S. Kedlaya, Semistable reduction for overconvergent 𝐹-isocrystals. II. A valuation-theoretic approach, Compos. Math. 144 (2008), no. 3, 657–672. MR 2422343, 10.1112/S0010437X07003296
  • 24. Kiran S. Kedlaya, Semistable reduction for overconvergent 𝐹-isocrystals. III. Local semistable reduction at monomial valuations, Compos. Math. 145 (2009), no. 1, 143–172. MR 2480498, 10.1112/S0010437X08003783
  • 25. K.S. Kedlaya, $ p$-adic differential equations, Cambridge Studies in Advanced Math. 125, Cambridge Univ. Press, Cambridge, 2010.
  • 26. K.S. Kedlaya, Good formal structures for flat meromorphic connections, I: Surfaces, Duke Math. J. 154 (2010), 343-418.
  • 27. K.S. Kedlaya, Semistable reduction for overconvergent $ F$-isocrystals, IV: Local semistable reduction at nonmonomial valuations, Compos. Math., to appear; arXiv preprint 0712.3400v4 (2010).
  • 28. Kiran S. Kedlaya and Liang Xiao, Differential modules on 𝑝-adic polyannuli, J. Inst. Math. Jussieu 9 (2010), no. 1, 155–201. MR 2576801, 10.1017/S1474748009000085
  • 29. Hagen Knaf and Franz-Viktor Kuhlmann, Abhyankar places admit local uniformization in any characteristic, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 6, 833–846 (English, with English and French summaries). MR 2216832, 10.1016/j.ansens.2005.09.001
  • 30. Hideyuki Majima, Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Mathematics, vol. 1075, Springer-Verlag, Berlin, 1984. MR 757897
  • 31. Bernard Malgrange, Connexions méromorphes. II. Le réseau canonique, Invent. Math. 124 (1996), no. 1-3, 367–387 (French). MR 1369422, 10.1007/s002220050057
  • 32. Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • 33. Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
  • 34. Takuro Mochizuki, Good formal structure for meromorphic flat connections on smooth projective surfaces, Algebraic analysis and around, Adv. Stud. Pure Math., vol. 54, Math. Soc. Japan, Tokyo, 2009, pp. 223–253. MR 2499558
  • 35. T. Mochizuki, Wild harmonic bundles and wild pure twistor $ D$-modules, arXiv preprint 0803.1344v3 (2009).
  • 36. Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 0308104
  • 37. Christel Rotthaus, Zur Komplettierung ausgezeichneter Ringe, Math. Ann. 253 (1980), no. 3, 213–226 (German). MR 597830, 10.1007/BF03219999
  • 38. Claude Sabbah, Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque 263 (2000), viii+190 (French, with English and French summaries). MR 1741802
  • 39. Michael Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), no. 2, 488–522. MR 2435647, 10.1016/j.aim.2008.05.006
  • 40. M. Temkin, Relative Riemann-Zariski spaces, Israel J. Math., to appear; arxiv preprint 0804.2843v1 (2008).
  • 41. M. Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: The non-embedded case, arXiv preprint 0904.1592v1 (2009).
  • 42. M. Temkin, Functorial desingularization over $ \mathbb{Q}$: boundaries and the embedded case, arXiv preprint 0912.2570v1 (2009).
  • 43. M. Temkin, Inseparable local uniformization, arXiv preprint 0804.1554v2 (2010).
  • 44. Michel Vaquié, Valuations, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 539–590 (French). MR 1748635
  • 45. V. S. Varadarajan, Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 1–42. MR 1339809, 10.1090/S0273-0979-96-00624-6
  • 46. Oscar Zariski, Local uniformization on algebraic varieties, Ann. of Math. (2) 41 (1940), 852–896. MR 0002864
  • 47. Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition; Graduate Texts in Mathematics, Vol. 29. MR 0389876

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Additional Information

Kiran S. Kedlaya
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: kedlaya@mit.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2010-00681-9
Received by editor(s): January 6, 2010
Received by editor(s) in revised form: July 14, 2010
Published electronically: September 17, 2010
Additional Notes: The author was supported by NSF CAREER grant DMS-0545904, DARPA grant HR0011-09-1-0048, MIT (NEC Fund, Green Career Development Professorship), IAS (NSF grant DMS-0635607, James D. Wolfensohn Fund).
Article copyright: © Copyright 2010 Kiran S. Kedlaya