Acta Mathematica

Extension theorems, non-vanishing and the existence of good minimal models

Abstract

We prove an extension theorem for effective purely log-terminal pairs (X, S + B) of non-negative Kodaira dimension ${\kappa (K_X+S+B)\ge 0}$ . The main new ingredient is a refinement of the Ohsawa–Takegoshi L2 extension theorem involving singular Hermitian metrics.

Note

The second author was partially supported by NSF research grant no. 0757897. During an important part of the preparation of this article, the third author was visiting KIAS (Seoul); he wishes to express his gratitude for the support and excellent working conditions provided by this institute. We would like to thank F. Ambro, B. Berndtsson, Y. Gongyo and C. Xu for interesting conversations about this article.

Article information

Source
Acta Math., Volume 210, Number 2 (2013), 203-259.

Dates
Revised: 4 November 2012
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892705

Digital Object Identifier
doi:10.1007/s11511-013-0094-x

Mathematical Reviews number (MathSciNet)
MR3070567

Zentralblatt MATH identifier
1278.14022

Rights

Citation

Demailly, Jean-Pierre; Hacon, Christopher D.; Păun, Mihai. Extension theorems, non-vanishing and the existence of good minimal models. Acta Math. 210 (2013), no. 2, 203--259. doi:10.1007/s11511-013-0094-x. https://projecteuclid.org/euclid.acta/1485892705

References

• Ambro F.: Nef dimension of minimal models. Math. Ann., 330, 309–322 (2004)
• Ambro F.: The moduli b-divisor of an lc-trivial fibration. Compos. Math., 141, 385–403 (2005)
• Berndtsson, B., The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman. Ann. Inst. Fourier (Grenoble), 46 (1996), 1083–1094.
• Berndtsson, B. & Păun, M., Quantitative extensions of pluricanonical forms and closed positive currents. Nagoya Math. J., 205 (2012), 25–65.
• Birkar C.: Ascending chain condition for log canonical thresholds and termination of log flips. Duke Math. J., 136, 173–180 (2007)
• Birkar C.: On existence of log minimal models II. J. Reine Angew. Math., 658, 99–113 (2011)
• Birkar, C., Cascini, P., Hacon, C.D. & McKernan, J., Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 23 (2010), 405–468.
• Claudon, B., Invariance for multiples of the twisted canonical bundle. Ann. Inst. Fourier (Grenoble), 57 (2007), 289–300.
• Corti, A. & Lazić, V., New outlook on the minimal model program, II. Preprint, 2010. arXiv:1005.0614 [math.AG].
• Demailly, J.-P., Singular Hermitian metrics on positive line bundles, in Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Math., 1507, pp. 87–104. Springer, Berlin–Heidelberg, 1992.
• Demailly, J.-P. On the Ohsawa–Takegoshi–Manivel L2 extension theorem, in Complex Analysis and Geometry (Paris, 1997), Progr. Math., 188, pp. 47–82. Birkhäuser, Basel, 2000.
• Demailly, J.-P. Analytic Methods in Algebraic Geometry. Surveys of Modern Mathematics, 1. Int. Press, Somerville, MA, 2012.
• Ein, L. & Popa,M., Extension of sections via adjoint ideals. Math. Ann., 352 (2012), 373–408.
• de Fernex, T. & Hacon, C.D., Deformations of canonical pairs and Fano varieties. J. Reine Angew. Math., 651 (2011), 97–126.
• Fujino, O., Abundance theorem for semi log canonical threefolds. Duke Math. J., 102 (2000), 513–532.
• Fujino, O. Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl., 35, pp. 63–75. Oxford Univ. Press, Oxford, 2007.
• Fujino, O. Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math. Sci., 47 (2011), 727–789.
• Fukuda, S., Tsuji’s numerically trivial fibrations and abundance. Far East J. Math. Sci. (FJMS), 5 (2002), 247–257.
• Gongyo, Y., Remarks on the non-vanishing conjecture. Preprint, 2012. arXiv:1201.1128 [math.AG].
• Gongyo, Y. & Lehmann, B., Reduction maps and minimal model theory. Preprint, 2011. arXiv:1103.1605 [math.AG].
• Hacon, C.D. & Kovács, S. J., Classification of Higher Dimensional Algebraic Varieties. Oberwolfach Seminars, 41. Birkhäuser, Basel, 2010.
• Hacon, C. D. & McKernan, J., Boundedness of pluricanonical maps of varieties of general type. Invent. Math., 166 (2006), 1–25.
• Hacon, C. D. & McKernan, J. Existence of minimal models for varieties of log general type. II. J. Amer. Math. Soc., 23 (2010), 469–490.
• Hacon, C. D., McKernan, J. & Xu, C., ACC for log canonical thresholds. Preprint, 2012. arXiv:1208.4150 [math.AG].
• Kawamata, Y., Pluricanonical systems on minimal algebraic varieties. Invent. Math., 79 (1985), 567–588.
• Kawamata, Y The Zariski decomposition of log-canonical divisors, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, ME, 1985), Proc. Sympos. Pure Math., 46, pp. 425–433. Amer. Math. Soc., Providence, RI, 1987.
• Kawamata, Y Abundance theorem for minimal threefolds. Invent. Math., 108 (1992), 229–246.
• Kawamata, Y On the cone of divisors of Calabi–Yau fiber spaces. Internat. J. Math., 8 (1997), 665–687.
• Keel, S., Matsuki, K. & McKernan, J., Log abundance theorem for threefolds. Duke Math. J., 75 (1994), 99–119.
• Klimek, M., Pluripotential Theory. London Mathematical Society Monographs, 6. Oxford University Press, New York, 1991.
• Kollár, J. & Mori, S., Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.
• Kollár, J. (ed.), Flips and Abundance for Algebraic Threefolds (Salt Lake City, UT, 1991). Astérisque, 211. Société Mathématique de France, Paris, 1992.
• Lai, C.-J., Varieties fibered by good minimal models. Math. Ann., 350 (2011), 533– 547.
• Lelong, P., Fonctions plurisousharmoniques et formes différentielles positives. Gordon & Breach, Paris, 1968.
• Lelong, P. Éléments extrémaux dans le côone des courants positifs fermés de type (1, 1) et fonctions plurisousharmoniques extrémales. C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A665–A667.
• Manivel, L., Un théorème de prolongement L2 de sections holomorphes d’un fibré hermitien. Math. Z., 212 (1993), 107–122.
• McNeal, J. D. & Varolin, D., Analytic inversion of adjunction: L2 extension theorems with gain. Ann. Inst. Fourier (Grenoble), 57 (2007), 703–718.
• Miyaoka, Y., Abundance conjecture for 3-folds: case v = 1. Compos. Math., 68 (1988), 203–220.
• Nakayama, N., Zariski-Decomposition and Abundance. MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004.
• Ohsawa, T., On the extension of L2 holomorphic functions. VI. A limiting case, in Explorations in Complex and Riemannian Geometry, Contemp. Math., 332, pp. 235–239. Amer. Math. Soc., Providence, RI, 2003.
• Ohsawa, T. Generalization of a precise L2 division theorem, in Complex Analysis in Several Variables, Adv. Stud. Pure Math., 42, pp. 249–261. Math. Soc. Japan, Tokyo, 2004.
• Ohsawa, T. & Takegoshi, K., On the extension of L2 holomorphic functions. Math. Z., 195 (1987), 197–204.
• Păun, M., Siu’s invariance of plurigenera: a one-tower proof. J. Differential Geom., 76 (2007), 485–493.
• Păun, M. Relative critical exponents, non-vanishing and metrics with minimal singularities. Invent. Math., 187 (2012), 195–258.
• Shokurov, V. V., Letters of a bi-rationalist. VII. Ordered termination. Tr. Mat. Inst. Steklova, 264 (2009), 184–208 (Russian); English translation in Proc. Steklov Inst. Math., 264 (2009), 178–200.
• Siu, Y.-T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 27 (1974), 53–156.
• Siu, Y.-T. Every Stein subvariety admits a Stein neighborhood. Invent. Math., 38 (1976/77), 89–100.
• Siu, Y.-T. Invariance of plurigenera. Invent. Math., 134 (1998), 661–673.
• 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), pp. 223–277. Springer, Berlin–Heidelberg, 2002.
• Siu, Y.-T. Finite generation of canonical ring by analytic method. Sci. China Ser. A, 51 (2008), 481–502.
• Siu, Y.-T. Abundance conjecture, in Geometry and Analysis. No. 2, Adv. Lect. Math., 18, pp. 271–317. Int. Press, Somerville, MA, 2011.
• Skoda, H., Application des techniques L2à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids. Ann. Sci. Éc. Norm. Super., 5 (1972), 545–579.
• Takayama S.: Pluricanonical systems on algebraic varieties of general type. Invent. Math., 165, 551–587 (2006)
• Takayama S.: On the invariance and the lower semi-continuity of plurigenera of algebraic varieties. J. Algebraic Geom., 16, 1–18 (2007)
• Tian G.: On Kähler–Einstein metrics on certain Kähler manifolds with C1(M) > 0. Invent. Math., 89, 225–246 (1987)
• Tsuji, H., Extension of log pluricanonical forms from subvarieties. Preprint, 2007. arXiv:0709.2710 [math.AG].
• Varolin D., Varolin D.: Takayama-type extension theorem. Compos. Math., 144, 522–540 (2008)