L 2 extension of adjoint line bundle sections
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1435-1477.

We prove an extension theorem of Ohsawa-Takegoshi type for line bundle sections on a subvariety of general codimension in a normal projective variety. Our method of proof gives conditions to be satisfied for such extension in a general setting, while such conditions are satisfied when the subvariety is given by an appropriate multiplier ideal sheaf.

Nous prouvons un théorème d’extension de type Ohsawa-Takegoshi pour les sections du fibré en droite de codimension générale dans une variété projective normale. Notre méthode donne des conditions qui doivent être satisfaites par de telles extensions dans un cadre général, alors qu’elles sont satisfaites quand la sous-variété est donnée par un faisceau d’idéaux multiplicateur approprié.

DOI: 10.5802/aif.2560
Classification: 32J25, 14E30
Keywords: $L^2$ extension, multiplier ideal sheaf, pluricanonical line bundle
Mot clés : Extension $L^2$, faisceau d’idéaux multiplicateur, fibré en droite pluricanonique
Kim, Dano 1

1 University of Chicago Dept. of Mathematics 5734 S. University Ave. Chicago, IL 60637 (USA)
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Kim, Dano. $L^2$ extension of adjoint line bundle sections. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1435-1477. doi : 10.5802/aif.2560. https://aif.centre-mersenne.org/articles/10.5802/aif.2560/

[1] Angehrn, Urban; Siu, Yum Tong Effective freeness and point separation for adjoint bundles, Invent. Math., Volume 122 (1995) no. 2, pp. 291-308 | DOI | MR | Zbl

[2] Berndtsson, Bo The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman, Ann. Inst. Fourier (Grenoble), Volume 46 (1996) no. 4, pp. 1083-1094 | DOI | Numdam | MR | Zbl

[3] Demailly, Jean-Pierre Complex analytic and differential geometry (book available at http://www-fourier.ujf-grenoble.fr/ demailly/books.html.)

[4] Demailly, Jean-Pierre L 2 vanishing theorems for positive line bundles and adjunction theory, Transcendental methods in algebraic geometry (Cetraro, 1994) (Lecture Notes in Math.), Volume 1646, Springer, Berlin, 1996, pp. 1-97 | DOI | MR | Zbl

[5] Demailly, Jean-Pierre On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry (Paris, 1997) (Progr. Math.), Volume 188, Birkhäuser, Basel, 2000, pp. 47-82 | MR | Zbl

[6] Fujino, Osamu Applications of Kawamata’s positivity theorem, Proc. Japan Acad. Ser. A Math. Sci., Volume 75 (1999) no. 6, pp. 75-79 http://projecteuclid.org/getRecord?id=euclid.pja/1148393905 | DOI | MR | Zbl

[7] Fujino, Osamu A Memorandum on the invariance of plurigenera, 2006 (private note)

[8] Grauert, Hans; Remmert, Reinhold Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 265, Springer-Verlag, Berlin, 1984 | MR | Zbl

[9] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978 (Pure and Applied Mathematics) | MR | Zbl

[10] Griffiths, Phillip A. Variations on a theorem of Abel, Invent. Math., Volume 35 (1976), pp. 321-390 | DOI | MR | Zbl

[11] Gunning, Robert C.; Rossi, Hugo Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs, N.J., 1965 | MR | Zbl

[12] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[13] Hörmander, Lars L 2 estimates and existence theorems for the ¯ operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[14] Kawamata, Yujiro On Fujita’s freeness conjecture for 3-folds and 4-folds, Math. Ann., Volume 308 (1997) no. 3, pp. 491-505 | DOI | MR | Zbl

[15] Kawamata, Yujiro Subadjunction of log canonical divisors. II, Amer. J. Math., Volume 120 (1998) no. 5, pp. 893-899 | DOI | MR | Zbl

[16] Kollár, János Singularities of pairs, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221-287 | Zbl

[17] Kollár, János Kodaira’s canonical bundle formula and adjunction, Flips for 3-folds and 4-folds (Oxford Lecture Ser. Math. Appl.), Volume 35, Oxford Univ. Press, Oxford, 2007, pp. 134-162 | DOI | MR

[18] Lazarsfeld, Robert Positivity in algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48-49, Springer-Verlag, Berlin, 2004 (Positivity for vector bundles, and multiplier ideals) | Zbl

[19] Lelong, P. Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1969 (Dunod, Paris) | Zbl

[20] Manivel, Laurent Un théorème de prolongement L 2 de sections holomorphes d’un fibré hermitien, Math. Z., Volume 212 (1993) no. 1, pp. 107-122 | DOI | MR | Zbl

[21] McNeal, Jeffery D. L 2 estimates on twisted Cauchy-Riemann complexes, 150 years of mathematics at Washington University in St. Louis (Contemp. Math.), Volume 395, Amer. Math. Soc., Providence, RI, 2006, pp. 83-103 | MR | Zbl

[22] McNeal, Jeffery D.; Varolin, Dror Analytic inversion of adjunction: L 2 extension theorems with gain, Ann. Inst. Fourier (Grenoble), Volume 57 (2007) no. 3, pp. 703-718 | DOI | Numdam | MR

[23] Ohsawa, Takeo On the extension of L 2 holomorphic functions. III. Negligible weights, Math. Z., Volume 219 (1995) no. 2, pp. 215-225 | DOI | MR | Zbl

[24] Ohsawa, Takeo; Takegoshi, Kenshō On the extension of L 2 holomorphic functions, Math. Z., Volume 195 (1987) no. 2, pp. 197-204 | DOI | MR | Zbl

[25] Păun, M. Siu’s Invariance of Plurigenera: a One-Tower Proof, J. Differential Geom., Volume 76 (2007) no. 3, pp. 485-493 | MR | Zbl

[26] Ransford, Thomas Potential theory in the complex plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[27] Rudin, Walter Functional analysis, International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991 | MR | Zbl

[28] Siu, Yum-Tong The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi, Geometric complex analysis (Hayama, 1995), World Sci. Publ., River Edge, NJ, 1996, pp. 577-592 | MR | Zbl

[29] Siu, Yum-Tong Invariance of plurigenera, Invent. Math., Volume 134 (1998) no. 3, pp. 661-673 | DOI | MR | Zbl

[30] Siu, Yum-Tong 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), Springer, Berlin, 2002, pp. 223-277 | MR | Zbl

[31] Skoda, Henri Application des techniques L 2 à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 545-579 | Numdam | MR | Zbl

[32] Takayama, Shigeharu Pluricanonical systems on algebraic varieties of general type, Invent. Math., Volume 165 (2006) no. 3, pp. 551-587 | DOI | MR | Zbl

[33] Varolin, Dror Analytic Methods in Algebraic Geometry, Lecture Notes, Stony Brook University, 2007

[34] Varolin, Dror A Takayama-type extension theorem, Compos. Math., Volume 144 (2008) no. 2, pp. 522-540 | DOI | MR | Zbl

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