Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-17T05:48:18.505Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction maps and minimal model theory

Part of: Birational geometry

Published online by Cambridge University Press:  04 December 2012

Yoshinori Gongyo  and
Brian Lehmann
Yoshinori Gongyo
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan (email: gongyo@ms.u-tokyo.ac.jp)
Brian Lehmann
Affiliation:
Department of Mathematics, Rice University, Houston, TX 77005, USA (email: blehmann@rice.edu)
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 use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.

Keywords

minimal modelabundance conjecturereduction map

MSC classification

Secondary: 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 2 , February 2013 , pp. 295 - 308
Copyright
© The Author(s) 2012

References

[Amb04]Ambro, F., Nef dimension of minimal models, Math. Ann. 330 (2004), 309–322.CrossRefGoogle Scholar
[Amb05]Ambro, F., The moduli b-divisor of an lc-trivial fibration, Compositio Math. 141 (2005), 385–403.CrossRefGoogle Scholar
[BCEK02+]Bauer, T., Campana, F., Eckl, T., Kebekus, S., Peternell, T., Rams, S., Szemberg, T. and Wotzlaw, L., A reduction map for nef line bundles, in Complex geometry (Springer, Berlin, 2002), 27–36.CrossRefGoogle Scholar
[Bir11]Birkar, C., On existence of log minimal models II, J. Reine Angew Math. 658 (2011), 99–113.Google Scholar
[BCHM10]Birkar, C., Cascini, P., Hacon, C. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.CrossRefGoogle Scholar
[Bou04]Boucksom, S., Divisorial Zariski decomposition on compact complex manifolds, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 45–76.CrossRefGoogle Scholar
[BBP09]Boucksom, S., Broustet, A. and Pacienza, G., Uniruledness of stable base loci of adjoint linear systems with and without Mori Theory, Preprint (2009), math.AG/0902.1142v2.Google Scholar
[BDPP04]Boucksom, S., Demailly, J.-P., Păun, M. and Peternell, T., The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., to appear, http://www-fourier.ujf-grenoble.fr/∼demailly/manuscripts/coneduality.pdf.Google Scholar
[CD11]Cacciola, S. and Di Biagio, L., Asymptotic base loci on singular varieties, Preprint (2011), math.AG/1105.1253.Google Scholar
[DHP10]Demailly, J.-P., Hacon, C. and Păun, M., Extension theorems, Non-vanishing and the existence of good minimal models, Preprint (2010), math.AG/1012.0493v1.Google Scholar
[Dru11]Druel, S., Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la premiére classe de Chern est nulle, Math. Z. 267 (2011), 413–423.CrossRefGoogle Scholar
[Eck04]Eckl, T., Tsuji’s numerical trivial fibrations, J. Algebraic Geom. 13 (2004), 617–639.CrossRefGoogle Scholar
[Eck05]Eckl, T., Numerically trivial foliations, Iitaka fibrations and the numerical dimension, Preprint (2005), math.AG/0508340.Google Scholar
[ELMNP06]Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M. and Popa, M., Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), 1701–1734.CrossRefGoogle Scholar
[Fuj11]Fujino, O., Semi-stable minimal model program for varieties with trivial canonical divisor, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 25–30.CrossRefGoogle Scholar
[FM00]Fujino, O. and Mori, S., A canonical bundle formula, J. Differential Geom. 56 (2000), 167–188.CrossRefGoogle Scholar
[Fuk02]Fukuda, S., Tsuji’s numerically trivial fibrations and abundance, Far East J. Math. Sci. (FJMS) 5 (2002), 247–257.Google Scholar
[Gon11]Gongyo, Y., On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero, Math. Res. Lett. 18 (2011), 991–1000.CrossRefGoogle Scholar
[Kaw85]Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), 567–588.CrossRefGoogle Scholar
[Kaw88]Kawamata, Y., Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), 93–163.CrossRefGoogle Scholar
[Kaw10]Kawamata, Y., On the abundance theorem in the case of ν=0, Amer. J. Math., to appear, math.AG/1002.2682.Google Scholar
[Lai11]Lai, C.-J., Varieties fibered by good minimal models, Math. Ann. 350 (2011), 533–547.CrossRefGoogle Scholar
[Leh11a]Lehmann, B., On Eckl’s pseudo-effective reduction map, Trans. Amer. Math. Soc., to appear, math.AG/1103.1073v1.Google Scholar
[Leh11b]Lehmann, B., Comparing numerical dimensions, Algebra Number Theory, to appear, math.AG/1103.0440v1.Google Scholar
[Nak04]Nakayama, N., Zariski decomposition and abundance, MSJ Memoirs, vol. 14 (Mathematical Society of Japan, Tokyo, 2004).CrossRefGoogle Scholar
[Siu11]Siu, Y. T., Abundance conjecture, in Geometry and analysis, Vol. 2, Advanced Lectures in Mathematics, vol. 18 (International Press, Somerville, MA, 2011), 271–317.Google Scholar
[Tak08]Takayama, S., On uniruled degenerations of algebraic varieties with trivial canonical divisor, Math. Z. 259 (2008), 487–501.CrossRefGoogle Scholar
[Tsu00]Tsuji, H., Numerically trivial fibration, Preprint (2000), math.AG/0001023.Google Scholar