Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T03:05:04.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantitative extensions of pluricanonical forms and closed positive currents

Published online by Cambridge University Press:  11 January 2016

Bo Berndtsson  and
Mihai Păun
Bo Berndtsson
Affiliation:
Chalmers University of Technology, SE-412 96 Gothenburg, [email protected]
Mihai Păun
Affiliation:
Institut Élie, Cartan de Nancy Université Henri Poincaré Nancy 1, B.P. 70239 54506 Vandocuvre-lès-Nancy, CEDEX [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish here several “invariance of plurigenera type” theorems for twisted pluricanonical forms and metrics of adjoint ℝ-bundles.

Keywords

32Q1514D0714E99
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 205 , March 2012 , pp. 25 - 65
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[1] Berndtsson, B., On the Ohsawa-Takegoshi extension theorem, Ann. Inst. Fourier (Grenoble) 46 (1996), 1083–1094.CrossRefGoogle Scholar
[2] Berndtsson, B., Integral formulas and the Ohsawa-Takegoshi extension theorem, Sci. China, Ser. A 48 (2005), 61–73.CrossRefGoogle Scholar
[3] Berndtsson, B. and Paun, M., Bergman kernels and the pseudo-effectivity of the relative canonical bundles, Duke Math. J. 145 (2008), 341–378.CrossRefGoogle Scholar
[4] Berndtsson, B. and Paun, M., A Bergman kernel proof of the Kawamata subadjunction theorem, preprint, arXiv:0804.3884v2 [math.AG] Google Scholar
[5] Boucksom, S., Cúones positifs des variétés complexes compactes, Ph.D. dissertation, Institut Fourier, Grenoble, France, 2002.Google Scholar
[6] Claudon, B., Invariance for multiples of the twisted canonical bundle, Ann. Inst. Fourier (Grenoble) 57 (2007), 289–300.CrossRefGoogle Scholar
[7] Demailly, J.-P., “Singular Hermitian metrics on positive line bundles” in Conference on Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Math. 1507, Springer, Berlin, 1992, 87–104.CrossRefGoogle Scholar
[8] Demailly, J.-P., Regularization of closed positive currents and intersection theory, J. Alge braic Geom. 1 (1992), 361–409.Google Scholar
[9] Demailly, J.-P., “On the Ohsawa-Takegoshi-Manivel extension theorem” in Conference on Complex Analysis and Geometry (Paris, 1997), Progr. Math. 188, Birkhauser, Basel, 1999, 47–82.CrossRefGoogle Scholar
[10] Demailly, J.-P., “Kähler manifolds and transcendental techniques in algebraic geometry” in International Congress of Mathematicians, I, Eur. Math. Soc, Zurich, 2007, 153–186.Google Scholar
[11] Demailly, J.-P., Analytic methods in algebraic geometry, preprint, 2009.Google Scholar
[12] Ein, L. and Popa, M., Extension of sections via adjoint ideals, preprint, arXiv:0811.4290 [math.AG] Google Scholar
[13] de Fernex, T. and Hacon, C. D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651 (2011), 97–126.Google Scholar
[14] de Fernex, T. and Hacon, C. D., Rigidity properties of Fano varieties, preprint, arXiv:0911.0504 [math.AG] Google Scholar
[15] Hacon, C. D., Extension theorems and the existence of flips, lecture series at Ober-wolfach Mathematical Institute, October 12–18, 2008.Google Scholar
[16] Hacon, C. D. and McKernan, J., Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1–25.CrossRefGoogle Scholar
[17] Hacon, C. D. and McKernan, J., “Extension theorems and the existence of flips” in Flips for 3-Folds and 4- Folds?, Oxford Lecture Ser. Math. Appl. 35, Oxford University Press, Oxford, 2007, 76–110.Google Scholar
[18] Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type, II, J. Amer. Math. Soc. 23 (2010), 469–490.CrossRefGoogle Scholar
[19] Kawamata, Y., Deformation of canonical singularities, J. Amer. Math. Soc. 12 (1999), 85–92.CrossRefGoogle Scholar
[20] Kawamata, Y., “On the extension problem of pluricanonical forms” in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1988), Contemp. Math. 241, Amer. Math. Soc., Providence, 1999, 193–207.CrossRefGoogle Scholar
[21] Kim, D., L2 extension of adjoint line bundle sections, Ann. Inst. Fourier (Grenoble) 60 (2010), 1435–1477.CrossRefGoogle Scholar
[22] Lazarsfeld, R., Positivity in Algebraic Geometry, Ergeb. Math. Grenzgeb. 48, Springer, Berlin, 2004.Google Scholar
[23] Levine, M., Pluri-canonical divisors on Kähler manifolds, Invent. Math. 74 (1983), 293–303.CrossRefGoogle Scholar
[24] Ohsawa, T., On the extension of L2 holomorphic functions, VI, A limiting case, Con-temp. Math. 332, Amer. Math. Soc., Providence, 2003, 235–239.Google Scholar
[25] Ohsawa, T., “Generalization of a precise L2 division theorem” in Complex Analysis in Several Variables (Kyoto, 2001), Adv. Stud. Pure Math. 42, Math. Soc. Japan, Tokyo, 2004, 249–261.Google Scholar
[26] Ohsawa, T. and Takegoshi, K., On the extension of L2 holomorphic functions, Math. Z. 195 (1987), 197–204.CrossRefGoogle Scholar
[27] Păaun, M., Siu’s invariance of plurigenera: A one-tower proof, J. Diff. Geom. 76 (2007), 485–493.Google Scholar
[28] Păaun, M., Relative critical exponents, non-vanishing and metrics with minimal singu-larities, preprint, arXiv:0807.3109 [math.AG] Google Scholar
[29] Siu, Y.-T., Invariance of plurigenera, Invent. Math. 134 (1998), 661–673.CrossRefGoogle Scholar
[30] 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 (Göttingen, 2000), Springer, Berlin, 2002, 223–277.CrossRefGoogle Scholar
[31] Takayama, S., Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551–587.CrossRefGoogle Scholar
[32] Takayama, S., On the invariance and lower semi-continuity of plurigenera of algebraic varieties, J. Algebraic Geom. 16 (2007), 1–18.CrossRefGoogle Scholar
[33] Tsuji, H., Extension of log pluricanonical forms from subvarieties, preprint, arXiv:0709.2710v2 [math.AG] Google Scholar
[34] Tsuji, H., Canonical singular Hermitian metrics on relative canonical bundles, preprint, arXiv:0704.0566v5 [math.AG] Google Scholar
[35] Tsuji, H., Canonical volume forms on compact Kähler manifolds, preprint, arXiv:0707.0111v1 [math.AG] Google Scholar
[36] Varolin, D., A Takayama-type extension theorem, Compos. Math. 144 (2008), 522–540.CrossRefGoogle Scholar