Advertisement

Science China Mathematics

, Volume 60, Issue 6, pp 967–976 | Cite as

Strong openness of multiplier ideal sheaves and optimal L 2 extension

  • Qi’An GuanEmail author
  • XiangYu Zhou
Articles

Abstract

In this paper, we reveal that our solution of Demailly’s strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal L 2 extension implies Berndtsson’s positivity of vector bundles associated to holomorphic fibrations over a unit disc.

Keywords

strong openness conjecture plurisubharmonic function multiplier ideal sheaf optimal L2 extension theorem 

MSC(2010)

32D15 32E10 32L10 32W05 32U05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berndtsson B. The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman. Ann Inst Fourier (Grenoble), 1996, 46: 1083–1094MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berndtsson B. Prekopa’s theorem and Kiselman’s minimal principle for plurisubharmonic functions. Math Ann, 1998, 312: 785–92MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berndtsson B. Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann Inst Fourier (Grenoble), 2006, 56: 1633–1662MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berndtsson B. Curvature of vector bundles associated to holomorphic fibrations. Ann of Math (2), 2009, 169: 531–560MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berndtsson B. The openness conjecture for plurisubharmonic functions. ArXiv:1305.5781, 2013Google Scholar
  6. 6.
    Blel M, Mimouni S K. Singularités et intégrabilité des fonctions plurisousharmoniques. Ann Inst Fourier (Grenoble), 2005, 55: 319–351MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boucksom S, Favre C, Jonsson M. Valuations and plurisubharmonic singularities. Publ Res Inst Math Sci. 2008, 44: 449–494MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cao J Y. Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact Kähler manifolds. Compos Math, 2014, 150: 1869–1902MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen B. Parameter dependence of the Bergman kernels. Adv Math, 2016, 299: 108–138MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demailly J P. On the Ohsawa-Takegoshi-Manivel L 2 extension theorem. In: Proceedings of the Conference in Honour of the 85th Birthday of Pierre Lelong. Complex Analysis and Geometry. Progress in Mathematics, vol, 188. Basel: Birkhäuser, 2000, 47–82zbMATHGoogle Scholar
  11. 11.
    Demailly J P. Multiplier ideal sheaves and analytic methods in algebraic geometry. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry. ICTP Lect Notes, vol. 6. Trieste: Abdus Salam Int Cent Theoret Phys, 2001, 1–148Google Scholar
  12. 12.
    Demailly J P. Analytic Methods in Algebraic Geometry. Beijing: Higher Education Press, 2010zbMATHGoogle Scholar
  13. 13.
    Demailly J P. Complex analytic and differential geometry. Electronically accessible at http://www-fourier.ujfgrenoble. fr/ demailly/books.htmlGoogle Scholar
  14. 14.
    Demailly J P, Ein L, Lazarsfeld R. A subadditivity property of multiplier ideals. Michigan Math J, 2000, 48: 137–156MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Demailly J P, Hacon C D, Păun M. Extension theorems, Non-vanishing and the existence of good minimal models. Acta Math, 2013, 210: 203–259MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Demailly J P, Kollár J. Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann Sci ´Ec Norm Supér (4), 2001, 34: 525–556CrossRefzbMATHGoogle Scholar
  17. 17.
    Favre C, Jonsson M. Valuative analysis of planar plurisubharmonic functions. Invent Math, 2005, 162: 271–311MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Favre C, Jonsson M. Valuations and multiplier ideals. J Amer Math Soc, 2005, 18: 655–684MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guan Q A, Zhou X Y. Optimal constant problem in the L 2 extension theorem. C R Math Acad Sci Paris, 2012, 350: 753–756MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guan Q A, Zhou X Y. Optimal constant in an L 2 extension problem and a proof of a conjecture of Ohsawa. Sci China Math, 2015, 58: 35–59MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guan Q A, Zhou X Y. A solution of an L 2 extension problem with an optimal estimate and applications. Ann of Math, 2015, 181: 1139–1208MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guan Q A, Zhou X Y. Strong openness conjecture for plurisubharmonic functions. ArXiv:1311.3781, 2013Google Scholar
  23. 23.
    Guan Q A, Zhou X Y. Strong openness conjecture and related problems for plurisubharmonic functions. ArXiv: 1401.7158, 2014Google Scholar
  24. 24.
    Guan Q A, Zhou X Y. A proof of Demailly’s strong openness conjecture. Ann of Math (2), 2015, 182: 605–616MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guan Q A, Zhou X Y. Effectiveness of Demailly’s strong openness conjecture and related problems. Invent Math, 2015, 202: 635–676MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Guan Q A, Zhou X Y. Characterization of multiplier ideal sheaves with weights of Lelong number one. Adv Math, 2015, 285: 1688–1705MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Guenancia H. Toric plurisubharmonic functions and analytic adjoint ideal sheaves. Math Z, 2012, 271: 1011–1035MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jonsson M, Mustată M. Valuations and asymptotic invariants for sequences of ideals. Ann Inst Fourier (Grenoble), 2012, 62: 2145–2209MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jonsson M, Mustată M. An algebraic approach to the openness conjecture of Demailly and Kollár. J Inst Math Jussieu, 2013, 13: 119–144CrossRefzbMATHGoogle Scholar
  30. 30.
    Kim D. The exactness of a general Skoda complex. Michigan Math J, 2014, 63: 3–18MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kiselman C O. Plurisubharmonic functions and potential theory in several complex variables. In: Development of Mathematics 1950–2000. Basel: BirkhÄauser, 2000, 655–714Google Scholar
  32. 32.
    Kollár J. Flips and Abundance for Algebraic Threefolds. Paris: Soc Math France, 1992zbMATHGoogle Scholar
  33. 33.
    Lazarsfeld R. Positivity in Algebraic Geometry, I: Classical Setting: Line Bundles and Linear Series; II: Positivity for Vector Bundles, and Multiplier Ideals. Berlin: Springer-Verlag, 2004zbMATHGoogle Scholar
  34. 34.
    Lehmann B. Algebraic bounds on analytic multiplier ideals. ArXiv:1109.4452v3, 2011Google Scholar
  35. 35.
    Manivel L. Un théorème de prolongement L 2 de sections holomorphes d’un fibré vectoriel. Math Z, 1993, 212: 107–122MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Matsumura S. An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. ArXiv:1308.2033, 2013Google Scholar
  37. 37.
    McNeal J, Varolin D. Analytic inversion of adjunction: L 2 extension theorems with gain. Ann Inst Fourier (Grenoble), 2007, 57: 703–718MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Nadel A. Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann of Math (2), 1990, 132: 549–596MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ohsawa T. On the extension of L 2 holomorphic functions. II. Publ Res Inst Math Sci, 1988, 24: 265–275MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ohsawa T. On the extension of L 2 holomorphic functions. III: Negligible weights. Math Z, 1995, 219: 215–225MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ohsawa T. On the extension of L 2 holomorphic functions. V: Effects of generalization. Nagoya Math J, 2001, 161: 1–21. Erratum to: “On the extension of L2 holomorphic functions. V: Effects of generalization” [Nagoya Math J, 2001, 161: 1–21]. Nagoya Math J, 2001, 163: 229MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ohsawa T. On the extension of L 2 holomorphic functions. VI: A limiting case. Contemp Math, 2003, 332: 235–239MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ohsawa T, Takegoshi K. On the extension of L 2 holomorphic functions. Math Z, 1987, 195: 197–204MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Păun M. Siu’s invariance of plurigenera: A one-tower proof. J Differential Geom, 2007, 76: 485–493MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sario L, Oikawa K. Capacity Functions. Die Grundlehren der mathematischen Wissenschaften. New York: Springer-Verlag, 1969zbMATHGoogle Scholar
  46. 46.
    Schiffer M, Spencer D C. Functionals of Finite Riemann Surfaces. Princeton: Princeton University Press, 1954zbMATHGoogle Scholar
  47. 47.
    Shokurov V. 3-fold log flips. Izv Ross Akad Nauk Ser Mat, 1992, 56: 105–203zbMATHGoogle Scholar
  48. 48.
    Sibony N. Quelques problèmes de prolongement de courants en analyse complexe (in French). Duke Math J, 1985, 52: 157–197MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Siu Y T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent Math, 1974, 27: 53–156MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Siu Y T. The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. In: Geometric Complex Analysis. Hayama: World Scientific, 1996, 577–592Google Scholar
  51. 51.
    Siu Y T. Invariance of plurigenera. Invent Math, 1998, 134: 661–673MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Siu Y T. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Complex Geometry. Berlin: Springer, 2002, 223–277CrossRefGoogle Scholar
  53. 53.
    Siu Y T. Multiplier ideal sheaves in complex and algebraic geometry. Sci China Ser A, 2005, 48: 1–31MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Siu Y T. Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds. In: Complex Analysis and Digital Geometry. Acta Univ Upsaliensis Skr Uppsala Univ C Organ Hist, vol. 86. Uppsala: Uppsala Universitet, 2009, 323–360Google Scholar
  55. 55.
    Skoda H. Sous-ensembles analytiques d’ordre fini ou infini dans Cn. Bull Soc Math France, 1972, 100: 353–408MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Straube E. Lectures on the L 2-Sobolev Theory of the \(\bar \partial \) -Neumann Problem. ESI Lectures in Mathematics and Physics. Zürich: Eur Math Soc, 2010zbMATHGoogle Scholar
  57. 57.
    Suita N. Capacities and kernels on Riemann surfaces. Arch Ration Mech Anal, 1972, 46: 212–217MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Tian G. On Kähler-Einstein metrics on certain Kähler manifolds with C 1(M) > 0. Invent Math, 1987, 89: 225–246MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yamada A. Topics related to reproducing kemels, theta functions and the Suita conjecture (in Japanese). Surikaisekikenkyusho Kokyuroku, 1998, 1067: 39–47zbMATHGoogle Scholar
  60. 60.
    Zhou X Y. A survey on L 2 extension problem. In: Complex Geometry and Dynamics. The Abel Symposium 2013. Cham: Springer, 2015, 291–307CrossRefGoogle Scholar
  61. 61.
    Zhu L F, Guan Q A, Zhou X Y. On the Ohsawa-Takegoshi L 2 extension theorem and the Bochner-Kodaira identity with non-smooth twist factor. J Math Pures Appl, 2012, 97: 579–601MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  3. 3.Institute of Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  4. 4.Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina

Personalised recommendations