Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-30T17:40:04.050Z 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)
Get access Rights & Permissions [Opens in a new window]

Abstract

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)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 2 , February 2013 , pp. 295 - 308
Copyright
© The Author(s) 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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