## Nagoya Mathematical Journal

### Quantitative extensions of pluricanonical forms and closed positive currents

#### Abstract

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

#### Article information

Source
Nagoya Math. J., Volume 205 (2012), 25-65.

Dates
First available in Project Euclid: 1 March 2012

https://projecteuclid.org/euclid.nmj/1330611001

Digital Object Identifier
doi:10.1215/00277630-1543778

Mathematical Reviews number (MathSciNet)
MR2891164

Zentralblatt MATH identifier
1248.32012

#### Citation

Berndtsson, Bo; Păun, Mihai. Quantitative extensions of pluricanonical forms and closed positive currents. Nagoya Math. J. 205 (2012), 25--65. doi:10.1215/00277630-1543778. https://projecteuclid.org/euclid.nmj/1330611001

#### References

• [1] B. Berndtsson, On the Ohsawa-Takegoshi extension theorem, Ann. Inst. Fourier (Grenoble) 46 (1996), 1083–1094.
• [2] B. Berndtsson, Integral formulas and the Ohsawa-Takegoshi extension theorem, Sci. China, Ser. A 48 (2005), 61–73.
• [3] B. Berndtsson and M. Păun, Bergman kernels and the pseudo-effectivity of the relative canonical bundles, Duke Math. J. 145 (2008), 341–378.
• [5] S. Boucksom, Cônes positifs des variétés complexes compactes, Ph.D. dissertation, Institut Fourier, Grenoble, France, 2002.
• [6] B. Claudon, Invariance for multiples of the twisted canonical bundle, Ann. Inst. Fourier (Grenoble) 57 (2007), 289–300.
• [7] J.-P. Demailly, “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.
• [8] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), 361–409.
• [9] J.-P. Demailly, “On the Ohsawa-Takegoshi-Manivel extension theorem” in Conference on Complex Analysis and Geometry (Paris, 1997), Progr. Math. 188, Birkhauser, Basel, 1999, 47–82.
• [10] J.-P. Demailly, “Kähler manifolds and transcendental techniques in algebraic geometry” in International Congress of Mathematicians, I, Eur. Math. Soc., Zurich, 2007, 153–186.
• [11] J.-P. Demailly, Analytic methods in algebraic geometry, preprint, 2009.
• [13] T. de Fernex and C. D. Hacon, Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651 (2011), 97–126.
• [15] C. D. Hacon, Extension theorems and the existence of flips, lecture series at Oberwolfach Mathematical Institute, October 12–18, 2008.
• [16] C. D. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1–25.
• [17] C. D. Hacon and J. McKernan, “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.
• [18] C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, II, J. Amer. Math. Soc. 23 (2010), 469–490.
• [19] Y. Kawamata, Deformation of canonical singularities, J. Amer. Math. Soc. 12 (1999), 85–92.
• [20] Y. Kawamata, “On the extension problem of pluricanonical forms” in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1988), Contemp. Math. 241, Amer. Math. Soc., Providence, 1999, 193–207.
• [21] D. Kim, L2 extension of adjoint line bundle sections, Ann. Inst. Fourier (Grenoble) 60 (2010), 1435–1477.
• [22] R. Lazarsfeld, Positivity in Algebraic Geometry, Ergeb. Math. Grenzgeb. 48, Springer, Berlin, 2004.
• [23] M. Levine, Pluri-canonical divisors on Kähler manifolds, Invent. Math. 74 (1983), 293–303.
• [24] T. Ohsawa, On the extension of L2 holomorphic functions, VI, A limiting case, Contemp. Math. 332, Amer. Math. Soc., Providence, 2003, 235–239.
• [25] T. Ohsawa, “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.
• [26] T. Ohsawa and K. Takegoshi, On the extension of L2 holomorphic functions, Math. Z. 195 (1987), 197–204.
• [27] M. Păun, Siu’s invariance of plurigenera: A one-tower proof, J. Diff. Geom. 76 (2007), 485–493.
• [29] Y.-T. Siu, Invariance of plurigenera, Invent. Math. 134 (1998), 661–673.
• [30] Y.-T. Siu, “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.
• [31] S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551–587.
• [32] S. Takayama, On the invariance and lower semi-continuity of plurigenera of algebraic varieties, J. Algebraic Geom. 16 (2007), 1–18.
• [36] D. Varolin, A Takayama-type extension theorem, Compos. Math. 144 (2008), 522–540.