Skip to main content
SpringerLink
Log in
Book cover

Mixed Twistor D-modules pp 103–139Cite as

Gluing of Good-KMS Smooth \(\mathcal{R}\)-Triples

  • Chapter

  • 1168 Accesses

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2125))

Abstract

Let \(X:={\bigl \{ (z_{1},\ldots,z_{n}) \in \mathbb{C}^{n}\,\big\vert \,\vert z_{i}\vert < 1\bigr \}}\), \(D_{i}:=\{ z_{i} = 0\} \cap X\) and \(D:=\bigcup _{ i=1}^{\ell}D_{i}\).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. C. Banica, Le complété formel d’un espace analytique le long d’un sous-espace: Un théoréme de comparaison. Manuscripta Math. 6, 207–244 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Beilinson, On the derived category of perverse sheaves, in K-Theory, Arithmetic and Geometry (Moscow, 1984–1986). Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), pp. 27–41

    Google Scholar 

  3. A. Beilinson, How to glue perverse sheaves, in K-Theory, Arithmetic and Geometry (Moscow, 1984–1986). Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), pp. 42–51

    Google Scholar 

  4. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981). Astérisque, vol. 100 (1982), pp. 5–171

    Google Scholar 

  5. J. Bingener, Über Formale Komplexe Räume. Manuscripta Math. 24, 253–293 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. J.-E. Björk, Analytic D-Modules and Applications (Kluwer Academic, Dordrecht, 1993)

    Book  MATH  Google Scholar 

  7. E. Cattani, A. Kaplan, Polarized mixed Hodge structures and the local monodromy of variation of Hodge structure. Invent. Math. 67, 101–115 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Cattani, A. Kaplan, W. Schmid, Degeneration of Hodge structures. Ann. Math. 123, 457–535 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Cattani, A. Kaplan, W. Schmid, L 2 and intersection cohomologies for a polarized variation of Hodge structure. Invent. Math. 87, 217–252 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. M.A. de Cataldo, L. Migliorini, The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup. (4) 38, 693–750 (2005)

    Google Scholar 

  11. P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Inst. Hautes Études Sci. Publ. Math. 35, 259–278 (1968)

    Article  MathSciNet  Google Scholar 

  12. P. Deligne, Équation Différentielles à Points Singuliers Réguliers. Lectures Notes in Mathematics, vol. 163 (Springer, Berlin, 1970)

    Google Scholar 

  13. P. Deligne, Cohomologie á supports propres, Théorie des Topos et Cohomologie Etale des Schémas, Springer Lecture Notes in Mathematics 305 (1973), 250–480

    Article  MathSciNet  Google Scholar 

  14. P. Deligne, Théorie de Hodge. I, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 (Gauthier-Villars, Paris, 1971), pp. 425–430

    Google Scholar 

  15. P. Deligne, Théorie de Hodge, II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Deligne, Théorie de Hodge, III. Inst. Hautes Études Sci. Publ. Math. 44, 5–77 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Deligne, B. Malgrange, J-P. Ramis, Singularités irrégulières, in Documents Mathématiques, vol. 5 (Société Mathématique de France, Paris, 2007)

    Google Scholar 

  18. A. Douady, Prolongement de faisceaux analytique cohérents (Travaux de Trautmann, Frisch-Guenot, Siu), in Seminaure Bourbaki 22e année, no. 366 (1969/1970)

    Google Scholar 

  19. H. Esnault, C. Sabbah, J.-D. Yu, E 1-Degeneration of the irregular Hodge filtration [arXiv:1302.4537] (2013)

    Google Scholar 

  20. H. Grauert, R. Remmert, Coherent Analytic Sheaves (Springer, Berlin, 1984)

    Book  MATH  Google Scholar 

  21. P.A. Griffiths, Hodge theory and geometry. Bull. Lond. Math. Soc. 36, 721–757 (2004)

    Article  MATH  Google Scholar 

  22. C. Hertling, tt ∗ geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555, 77–161 (2003)

    Google Scholar 

  23. C. Hertling, C. Sevenheck, Nilpotent orbits of a generalization of Hodge structures. J. Reine Angew. Math. 609, 23–80 (2007)

    MathSciNet  MATH  Google Scholar 

  24. C. Hertling, C. Sevenheck, Limits of families of Brieskorn lattices and compactified classifying spaces. Adv. Math. 223, 1155–1224 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. C. Hertling, C. Sevenheck, Twistor structures, tt∗-geometry and singularity theory, in From Hodge Theory to Integrability and TQFT tt ∗ -Geometry. Proceedings of Symposia in Pure Mathematics, vol.78 (American Mathematical Society, Providence, 2008), pp. 49–73

    Google Scholar 

  26. N. Hitchin, The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Hotta, K. Takeuchi, T. Tanisaki, D-Modules, Perverse Sheaves, and Representation Theory. Progress in Mathematics, vol. 236 (Birkhäuser, Boston, 2008)

    Google Scholar 

  28. M. Kashiwara, On the maximally overdetermined system of linear differential equations, I. Publ. Res. Inst. Math. Sci. 10, 563–579 (1974/1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, in Algebraic Geometry (Tokyo/Kyoto, 1982). Lecture Notes in Mathematics, vol. 1016 (Springer, Berlin, 1983), pp. 134–142

    Google Scholar 

  30. M. Kashiwara, The Riemann-Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 20, 319–365 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Kashiwara, The asymptotic behavior of a variation of polarized Hodge structure. Publ. Res. Inst. Math. Sci. 21, 853–875 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Kashiwara, A study of variation of mixed Hodge structure. Publ. Res. Inst. Math. Sci. 22, 991–1024 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Kashiwara, Regular holonomic D-modules and distributions on complex manifolds. in Complex Analytic Singularities. Advanced Studies in Pure Mathematics, vol. 8 (North-Holland, Amsterdam, 1987), pp. 199–206

    Google Scholar 

  34. M. Kashiwara, D-Modules and Microlocal Calculus. Translations of Mathematical Monographs. Iwanami Series in Modern Mathematics, vol. 217 (American Mathematical Society, Providence, 2003)

    Google Scholar 

  35. M. Kashiwara, T. Kawai, The Poincaré lemma for variations of polarized Hodge structure. Publ. Res. Inst. Math. Sci. 23, 345–407 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Kashiwara, P. Schapira, Sheaves on Manifolds (Springer, Berlin, 1990)

    Book  MATH  Google Scholar 

  37. L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, in From Hodge Theory to Integrability and TQFT tt*-Geometry. Proceedings of Symposia in Pure Mathematics, vol. 78 (American Mathematical Society, Providence, 2008), pp. 87–174

    Google Scholar 

  38. K. Kedlaya, Good formal structures for flat meromorphic connections, I: surfaces. Duke Math. J. 154, 343–418 (2010)

    MathSciNet  MATH  Google Scholar 

  39. K. Kedlaya, Good formal structures for flat meromorphic connections, II: excellent schemes. J. Am. Math. Soc. 24, 183–229 (2011)

    MathSciNet  MATH  Google Scholar 

  40. R. MacPherson, K. Vilonen, Elementary construction of perverse sheaves. Invent. Math. 84, 403–435 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  41. B. Malgrange, Ideals of differentiable functions, in Tata Institute of Fundamental Research Studies in Mathematics, vol. 3 (Tata Institute of Fundamental Research/Oxford University Press, Bombay/London, 1967)

    Google Scholar 

  42. B. Malgrange, Polynômes de Bernstein-Sato et cohomologie évanescente, in Analysis and Topology on Singular Spaces, II, III (Luminy, 1981). Astérisque, vol. 101–102 (1983), pp. 243–267

    Google Scholar 

  43. B. Malgrange, Équations Différentielles à Coefficients Polynomiaux. Progress in Mathematics, vol. 96 (Birkhäuser, Boston, 1991)

    Google Scholar 

  44. B. Malgrange, Connexions méromorphes, II. Le réseau canonique. Invent. Math. 124, 367–387 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Malgrange, On irregular holonomic D-modules, in Éléments de la Théorie des Systèmes Différentiels Géométriques. Sémin. Congr., vol. 8 (Société Mathématique de France, Paris, 2004), pp. 391–410

    Google Scholar 

  46. Z. Mebkhout, Une équivalence de catégories. Compos. Math. 51, 51–62 (1984)

    MathSciNet  MATH  Google Scholar 

  47. Z. Mebkhout, Une autre équivalence de catégories. Compos. Math. 51, 63–88 (1984)

    MathSciNet  Google Scholar 

  48. Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les D X -modules cohérents. With supplementary material by the author and L. Narváez Macarro. Travaux en Cours [Works in Progress], vol. 35 (Hermann, Paris, 1989), pp. x+254

    Google Scholar 

  49. S. Mizohata, The Theory of Partial Differential Equations (Translated from the Japanese by Katsumi Miyahara) (Cambridge University Press, New York, 1973)

    Google Scholar 

  50. T. Mochizuki, Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure. J. Diff. Geom. 62, 351–559 (2002)

    MathSciNet  MATH  Google Scholar 

  51. T. Mochizuki, Kobayashi-Hitchin correspondence for Tame harmoinc bundles and an application. Astérisque 309 viii+117 pp (2006)

    Google Scholar 

  52. T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules I, II. Mem. Am. Math. Soc. 185 vii+565 pp (2007)

    Google Scholar 

  53. T. Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles II. Geom. Topol. 13, 359–455 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  54. T. Mochizuki, Good formal structure for meromorphic flat connections on smooth projective surfaces, in Algebraic Analysis and Around. Advanced Studies in Pure Mathematics, Math. Soc. Japan, Tokyo, vol. 54 (2009), pp. 223–253

    Google Scholar 

  55. T. Mochizuki, Wild harmonic bundles and wild pure twistor D-modules. Astérisque 340 x+607 pp (2011)

    Google Scholar 

  56. T. Mochizuki, Asymptotic behaviour of variation of pure polarized TERP structures. Publ. Res. Inst. Math. Sci. 47, 419–534 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. T. Mochizuki, Holonomic \(\mathcal{D}\) -Module with Betti Structure. Mémoire de la SMF, vol. 138–139 (Société Mathématique de France, Paris, 2014)

    Google Scholar 

  58. T. Mochizuki, The Stokes structure of good meromorphic flat bundle. J. Inst. Math. Jussieu 10, 675–712 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Mochizuki, A twistor approach to the Kontsevich complexes [arXiv:1501.04145] (2015)

    Google Scholar 

  60. T. Mochizuki, Twistor property of GKZ-hypergeometric systems [arXiv:1501.04146] (2015)

    Google Scholar 

  61. T. Mochizuki, On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles, Ann. Inst. Fourier (Grenoble) 59, 2819–2837 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  62. C. Peters, J. Steenbrink, Mixed Hodge Structure (Springer, Berlin, 2008)

    Google Scholar 

  63. W. Rudin, Functional analysis, in International Series in Pure and Applied Mathematics, 2nd edn. (McGraw-Hill, New York, 1991)

    Google Scholar 

  64. C. Sabbah, Vanishing cycles and Hermitian duality. Proc. Steklov Inst. Math. 238, 194–214 (2002)

    MathSciNet  Google Scholar 

  65. C. Sabbah, Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2. Astérisque 263 viii+190 pp (2000)

    Google Scholar 

  66. C. Sabbah, Polarizable twistor D-modules. Astérisque 300 vi+208 pp (2005)

    Google Scholar 

  67. C. Sabbah, Wild twistor D-modules, in Algebraic Analysis and Around. Advanced Studies in Pure Mathematics, vol. 54 (Mathematical Society of Japan, Tokyo, 2009), pp. 293–353

    Google Scholar 

  68. C. Sabbah, Introduction to Stokes Structures. Lecture Notes in Mathematics, vol. 2060 (Springer, Heidelberg, 2013)

    Google Scholar 

  69. M. Saito, Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24, 849–995 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  70. M. Saito, Duality for vanishing cycle functors. Publ. Res. Inst. Math. Sci. 25, 889–921 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  71. M. Saito, Induced D-modules and differential complexes. Bull. Soc. Math. Fr. 117, 361–387 (1989)

    MATH  Google Scholar 

  72. M. Saito, Introduction to mixed Hodge modules, Actes du Colloque de Théorie de Hodge (Luminy, 1987). Astérisque No. 179–180 10, 145–162 (1989)

    Google Scholar 

  73. M. Saito, Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26, 221–333 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  74. M. Saito, Mixed Hodge modules and applications, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 725–734, Math. Soc. Japan, Tokyo, 1991

    Google Scholar 

  75. M. Saito, On the theory of mixed Hodge modules, Selected papers on number theory, algebraic geometry, and differential geometry, 47–61, Amer. Math. Soc. Transl. Ser. 2, 160, Amer. Math. Soc., Providence, RI, 1994

    Google Scholar 

  76. M. Saito, Appendix to H. Esnault, C. Sabbah, J.-D. Yu, E 1-Degeneration of the irregular Hodge filtration [arXiv:1302.4537]

    Google Scholar 

  77. W. Schmid, Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  78. C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)

    Article  MATH  Google Scholar 

  79. C. Simpson, Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3(3), 713–770 (1990)

    Article  MATH  Google Scholar 

  80. C. Simpson, Higgs bundles and local systems. Publ. I.H.E.S. 75, 5–95 (1992)

    Google Scholar 

  81. C. Simpson, Some families of local systems over smooth projective varieties. Ann. Math. 138(2), 337–425 (1993)

    Article  MATH  Google Scholar 

  82. C. Simpson, Mixed twistor structures [math.AG/9705006] (1997)

    Google Scholar 

  83. C. Simpson, The Hodge filtration on nonabelian cohomology, in Algebraic geometry—Santa Cruz 1995. Proceedings of the Symposia Pure Mathematics, vol. 62, Part 2 (American Mathematical Society, Providence, 1997), pp. 217–281

    Google Scholar 

  84. C. Simpson, Local systems on proper algebraic V-manifolds. Pure Appl. Math. Q. 7, 1675–1759 (2011) [Special Issue: In memory of Eckart Viehweg]

    Google Scholar 

  85. Y. T. Siu, Techniques of Extension of Analytic Objects, vol. 8 (Marcel Dekker, New York, 1974)

    MATH  Google Scholar 

  86. J. Steenbrink, Limits of Hodge structures. Invent. Math. 31, 229–257 (1975/1976)

    Article  MathSciNet  MATH  Google Scholar 

  87. J. Steenbrink, S. Zucker, Variation of mixed Hodge structure, I. Invent. Math. 80, 489–542 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  88. S. Zucker, Hodge theory with degenerating coefficients: L 2 cohomology in the Poincaré metric. Ann. Math. (2) 109, 415–476 (1979)

    Google Scholar 

  89. S. Zucker, Variation of mixed Hodge structure, II. Invent. Math. 80, 543–565 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  90. J-L. Verdier, Extension of a perverse sheaf over a closed subspace, in Differential Systems and Singularities (Luminy, 1983). Astérisque, vol. 130 (1985), pp. 210–217

    Google Scholar 

  91. J. Włodarczyk, Resolution of singularities of analytic spaces, in Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova (2009), pp. 31–63

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kyoto, Japan

    Takuro Mochizuki

Authors
  1. Takuro Mochizuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Mochizuki, T. (2015). Gluing of Good-KMS Smooth \(\mathcal{R}\)-Triples. In: Mixed Twistor D-modules. Lecture Notes in Mathematics, vol 2125. Springer, Cham. https://doi.org/10.1007/978-3-319-10088-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation