Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-29T19:50:59.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-dimensional foliated Mori theory

Part of: Birational geometry Complex dynamical systems

Published online by Cambridge University Press:  14 November 2019

Calum Spicer
Calum Spicer*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email calum.spicer@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of $K_{{\mathcal{F}}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.

Keywords

foliationsminimal model programKleiman–Mori conefoliation singularities

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 37F75: Holomorphic foliations and vector fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 1 , January 2020 , pp. 1 - 38
Copyright
© The Author 2019 

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.)

Footnotes

1

Current address: Department of Mathematics, King’s College London, London WC2R 2LS, UK

References

Ambro, F., Non-klt techniques , in Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and its Applications, vol. 35 (Oxford University Press, Oxford, 2007), 163–170.Google Scholar
Araujo, C. and Druel, S., On Fano foliations , Adv. Math. 238 (2013), 70–118.Google Scholar
Araujo, C. and Druel, S., On codimension 1 del Pezzo foliations on varieties with mild singularities , Math. Ann. 360 (2014), 769–798.Google Scholar
Artin, M., Algebraization of formal moduli: II , Ann. of Math. (2) 91 (1970), 88–135.Google Scholar
Beltrametti, M. C. and Sommese, A. J., The adjunction theory of complex projective varieties (Walter de Gruyter, 1995).Google Scholar
Bogomolov, F. and McQuillan, M., Rational curves on foliated varieties, Preprint (2001), Institut des Hautes Études Scientifiques.Google Scholar
Brunella, M., Minimal models of foliated algebraic surfaces , Bull. Soc. Math. France 127 (1999), 289–305.Google Scholar
Brunella, M., Birational geometry of foliations , in Monografías de Matemática (Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2000).Google Scholar
Brunella, M. and Perrone, C., Exceptional singularities of codimension one holomorphic foliations , Publ. Mat. 55 (2011), 295–312.Google Scholar
Cacciola, S. and Lopez, A. F., Nakamaye’s theorem on log canonical pairs , Ann. Inst. Fourier (Grenoble) 64 (2014), 2283–2298.Google Scholar
Cano, F., Reduction of the singularities of codimension one singular foliations in dimension three , Ann. of Math. (2) 160 (2004), 907–1011.Google Scholar
Cano, F. and Cerveau, D., Desingularization of nondicritical holomorphic foliations and existence of separatrices , Acta Math. 169 (1992), 1–103.Google Scholar
Cerveau, D. and Lins Neto, A, Frobenius theorem for foliations on singular varieties , Bull. Braz. Math. Soc. (N.S.) 39 (2008), 447–469.Google Scholar
Fujino, O., Fundamental theorems for the log minimal model program , Publ. Res. Inst. Math. Sci. 47 (2011), 727–789.Google Scholar
Greb, D., Kebekus, S., Kovács, S. J. and Peternell, T., Differential forms on log canonical spaces , Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87–169.Google Scholar
Greb, D., Kebekus, S. and Peternell, T., Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties , Duke Math. J. 165 (2016), 1965–2004.Google Scholar
Hartshorne, R., Cohomological dimension of algebraic varieties , Ann. of Math. (2) 88 (1968), 403–450.Google Scholar
Hartshorne, R., Algebraic geometry (Springer, New York, 1977).Google Scholar
Kawamata, Y., Subadjunction of log canonical divisors. II , Amer. J. Math. 120 (1998), 893–899.Google Scholar
Keel, S., Matsuki, K. and McKernan, J., Log abundance theorem for threefolds , Duke Math. J. 75 (1994), 99–119.Google Scholar
Kollár, J., Extremal rays on smooth threefolds , Ann. Sci. Éc. Norm. Supér. (4) 24 (1991), 339–361.Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).Google Scholar
Loray, F., Pereira, J. and Touzet, F., Singular foliations with trivial canonical class , Invent. Math. 213 (2018), 1327–1380.Google Scholar
Matsuki, K., Introduction to the Mori program, Universitext (Springer, New York, NY, 2002).Google Scholar
McQuillan, M., Semi-stable reduction of foliations, Preprint (2005), Institut des Hautes Études Scientifiques, IHES/M/05/02.Google Scholar
McQuillan, M., Canonical models of foliations , Pure Appl. Math. Q. 4 (2008), 877–1012.Google Scholar
Miyaoka, Y., Deformations of a morphism along a foliation and applications , in Algebraic Geometry – Bowdoin 1985, Part 1, Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 245–268.Google Scholar
Moerdijk, I. and Mrčun, J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91 (Cambridge University Press, Cambridge, 2003).Google Scholar
Nakayama, N., The lower semicontinuity of the plurigenera of complex varieties , in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 551–590.Google Scholar
Rockafellar, R. T., Convex analysis, Princeton Mathematical Series, vol. 28 (Princeton University Press, Princeton, NJ, 1970).Google Scholar
Shepherd-Barron, N., Miyaoka’s theorems on the generic seminegativity of T X and on the kodaira dimension of minimal regular threefolds , in Flips and abundance for algebraic threefolds ed. Kollár, J. (Société Mathématique de France, 1992), 103–114.Google Scholar
Touzet, F., Feuilletages holomorphes de codimension un dont la classe canonique est triviale , Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 655–668.Google Scholar