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Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles

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Abstract

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin’s application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the \(\bar \partial \) equation with cut-off functions and additional blowup weight functions.

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References

  1. Andreotti A, Vesentini E. Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Inst Hautes Études Sci Publ Math, 1965, 25: 81–130

    Article  MathSciNet  Google Scholar 

  2. Angehrn U, Siu Y T. Effective freeness and point separation for adjoint bundles. Invent Math, 1995, 122: 291–308

    Article  MathSciNet  MATH  Google Scholar 

  3. Artin M. On the Solutions of Analytic Equations. Invent Math, 1968, 5: 277–291

    Article  MathSciNet  MATH  Google Scholar 

  4. Artin M. Algebraic approximation of structures over complete local rings. Pub Math IHES, 1969, 36: 23–58

    MathSciNet  MATH  Google Scholar 

  5. Berndtsson B. The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman. Ann Inst Fourier (Grenoble), 1996, 46: 1083–1094

    MathSciNet  MATH  Google Scholar 

  6. Bombieri E. Algebraic values of meromorphic maps. Invent Math, 1970, 10: 267–287, Addendum Invent Math, 1970, 11: 163–166

    Article  MathSciNet  MATH  Google Scholar 

  7. Bombieri E, Lang E. Analytic subgroups of group varieties, Invent Math, 1970, 11: 1–14

    Article  MathSciNet  MATH  Google Scholar 

  8. Bombieri E, Pila J. The number of integral points on arcs and ovals. Duke Math J, 1989, 59: 337–357

    Article  MathSciNet  MATH  Google Scholar 

  9. Brieskorn E. Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta Math, 1970, 2: 103–161

    Article  MathSciNet  MATH  Google Scholar 

  10. Budur N. Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers. Adv Math, 2009, 221: 217–250

    Article  MathSciNet  MATH  Google Scholar 

  11. Campana F, Peternell T, Toma M. Geometric stability of the cotangent bundle and the universal cover of a projective manifold. arXiv:math/0405093

  12. Demailly J P. Champs magnétiques et inégalités de Morse pour la d″-cohomologie. Ann Inst Fourier (Grenoble), 1985, 35: 189–229

    MathSciNet  MATH  Google Scholar 

  13. Donnelly H, Fefferma C. L 2-cohomology and index theorem for the Bergman metric. Ann of Math, 1983, 118: 593–618

    Article  MathSciNet  MATH  Google Scholar 

  14. Donnelly H, Xavier F. On the differential form spectrum of negatively curved Riemannian manifolds. Amer J Math, 1984, 106: 169–185

    Article  MathSciNet  MATH  Google Scholar 

  15. Gelfond A O. Sur le septième Problème de D. Hilbert. Comptes Rendus Acad Sci URSS Moscou, 1934, 2: 1–6, Bull Acad Sci URSS Leningrade, 1934, 7: 623–634

    MathSciNet  Google Scholar 

  16. Hensel K. Über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1897, 6: 83–88

    Google Scholar 

  17. Hilbert D. Mathematische Probleme. Nachr Königl Ges der Wiss zu Göttingen, Math Phys Klasse, 1900, 251–297

  18. Hörmander L. L 2 estimates and existence theorems for the \(\bar \partial \) operator. Acta Math, 1965, 113}: 89–1

  19. Kodaira K. On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann of Math, 1954, 60: 28–48

    Article  MathSciNet  MATH  Google Scholar 

  20. Kim D. L 2 extension of adjoint line bundle sections. Ann Inst Fourier (Grenoble), 2010, 60: 1435–1477

    MathSciNet  MATH  Google Scholar 

  21. Kohn J J. Harmonic integrals on strongly pseudo-convex manifolds I. Ann of Math, 1963, 78: 112–148; II. Ann of Math, 1964, 79: 450–472

    Article  MathSciNet  MATH  Google Scholar 

  22. Lang S. Transcendental points on group varieties. Topology, 1962, 1: 313–318

    Article  MathSciNet  MATH  Google Scholar 

  23. Lang S. Algebraic values of meromorphic functions. Topology, 1965, 3: 183–191

    Article  MathSciNet  MATH  Google Scholar 

  24. Lang S. Introduction to transcendental numbers. Addison-Wesley Publishing Co Reading, Mass-London-Don Mills, Ont 1966

    MATH  Google Scholar 

  25. Manivel L. Un théorème de prolongement L 2 de sections holomorphes d’un fibré hermitien. Math Z, 1993, 212: 107–122

    Article  MathSciNet  MATH  Google Scholar 

  26. Milnor J. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61 Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo 1968

  27. Morrey Jr C B. The analytic embedding of abstract real-analytic manifolds. Ann of Math, 1958, 68: 159–201

    Article  MathSciNet  MATH  Google Scholar 

  28. Nevanlinna R. Zur Theorie der Meromorphen Funktionen. Acta Math, 1925, 46: 1–99

    Article  MathSciNet  MATH  Google Scholar 

  29. Ohsawa T, Takegoshi K. On the extension of L 2 holomorphic functions. Math Zeitschr, 1987, 195: 197–204

    Article  MathSciNet  MATH  Google Scholar 

  30. Paun M. Siu’s invariance of plurigenera: a one-tower proof. J Differential Geom, 2007, 76: 485–493

    MathSciNet  MATH  Google Scholar 

  31. Pila J, Wilkie A J. The rational points of a definable set. Duke Math J, 2006, 33: 591–616

    Article  MathSciNet  Google Scholar 

  32. Popovici D. L 2 extension for jets of holomorphic sections of a Hermitian line bundle. Nagoya Math J, 2005, 180: 1–34

    MathSciNet  MATH  Google Scholar 

  33. Schneider T. Transzendenzuntersuchungen periodischer Funktionen I, II. J Reine Angew Math, 1934, 172: 65–74

    MATH  Google Scholar 

  34. Simpson C. Subspaces of moduli spaces of rank one local systems. Ann Sci École Norm Sup, 1993, 26: 361–401

    MathSciNet  MATH  Google Scholar 

  35. Siu Y T. The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. Geometric complex analysis (Hayama, 1995), 577–592, River Edge, NJ: World Sci Publ, 1996

    Google Scholar 

  36. Siu Y T. Invariance of plurigenera. Invent Math, 1998, 134: 661–673

    Article  MathSciNet  MATH  Google Scholar 

  37. Siu Y T. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. Complex geometry (Göttingen, 2000), 223–277, Springer, Berlin, 2002

    Chapter  Google Scholar 

  38. Siu Y T. Abundance Conjecture in Geometry and Analysis, Vol II. ed. Lizhen Ji, International Press 2010, pp.271–317. (arXiv:math/0912.0576)

  39. Takayama S. Pluricanonical systems on algebraic varieties of general type. Invent Math, 2006, 165: 551–587

    Article  MathSciNet  MATH  Google Scholar 

  40. Varolin D. A Takayama-type extension theorem. Compos Math, 2008, 144: 522–540

    Article  MathSciNet  MATH  Google Scholar 

  41. Wavrik J J. A theorem on solutions of analytic equations with applications to deformations of complex structures. Math Ann, 1975, 216: 127–142

    Article  MathSciNet  MATH  Google Scholar 

  42. Weil A. Introduction à l’étude des variétés kählériennes. Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci Ind no 1267, Hermann, Paris 1958

    Google Scholar 

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Correspondence to Yum-Tong Siu.

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Dedicated to Fabrizio Catanese on the Occasion of his 60th Birthday

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Siu, YT. Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles. Sci. China Math. 54, 1767–1802 (2011). https://doi.org/10.1007/s11425-011-4293-7

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