An Introduction to Non-Commutative Mori Theory

  • Michael McQuillan
Part of the Progress in Mathematics book series (PM, volume 202)


Abstract. The classification of algebraic surfaces of Enriques and Kodaira, and its conjectured higher dimensional generalisation by S. Mori et al, leaves as many questions unanswered as it solves; for example: which surfaces are hyperbolic? The highly conceptual nature of Mori’s programme permits its extension to the birational classification of ordinary differential equations, and this in its full generality is the proper definition of non-commutative Mori theory. A remarkable observation of [2] is that not only can much of Mori’s rational curve technology be brought to bear on this problem, but that many of the basic theorems of classification depend not on the dimension of the space on which the ODE is defined, but rather the dimension of its solutions. This suggests that, despite the difficulty of Mori’s programme for higher dimensional varieties, its analogue for foliations by curves in any dimension will succeed. Whence a conjectural classification for ODEs on surfaces, and an exhaustive, yet wholly algebraic understanding of their hyperbolicity by way of the method of dynamic Diophantine approximation. A refined version of the programme leads similarly to the analogue of the Mordell conjecture for algebraic surfaces over function fields, which in [13] it had already succeeded in establishing under the hypothesis of positive index. Innovation in Diophantine approximation would lead to its applicability in arithmetic per se. Other applications are anticipated.


Abelian Variety Mori Theory Algebraic Surface Diophantine Approximation Kodaira Dimension 
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  1. [1]
    E. Arrondo, I. Sols and B. Speiser. ‘Global moduli for contacts’ Ark. Math. 35 (1997).Google Scholar
  2. [2]
    F. A. Bogomolov and M. McQuillan. ‘Rational curves on foliated varieties’ IHES pre-print to appear Oct. 2000.Google Scholar
  3. [3]
    M. Brunella. ‘Minimal models of foliated algebraic surfaces’ Bull. SMF 127 (1999) 289–305.MathSciNetzbMATHGoogle Scholar
  4. [4]
    M. Brunella. ‘Courbes entières et feuielletages holomorphes’ Enseign. Math. 45 (1999) 195–216.MathSciNetzbMATHGoogle Scholar
  5. [5]
    A. Connes. ‘Non-commutative geometry’ Academic Press 1994.Google Scholar
  6. [6]
    J.-P. Demailly. ‘Algebraic criteria for Kobayashi hyperbolic varieties and jet differentials’ Proc. Symp. Math. 62 (1997).Google Scholar
  7. [7]
    G. Faltings. ‘Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpen’ Invent. Math. 73 (1983) 349–366.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G. Faltings. ‘Diophantine approximation on abelian varieties’ Ann. Math. 133 (1991) 549–576.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    R. Friedman and Z. Qin. The smooth invariance of the Kodaira dimension of a complex surface Math. Res. Lett. 1 (1994) 369–376.MathSciNetzbMATHGoogle Scholar
  10. [10]
    J. Kollár et al. ‘Flips and abundance for algebraic Threefolds’ Astérique 211 (1992).Google Scholar
  11. [11]
    S. Lang. ‘Number Theory III’ Springer-Verlag 1991.Google Scholar
  12. [12]
    M. McQuillan. ‘Diophantine approximation and foliations’ Inst. Hautes Études Sci. Publ. Math. 87 (1999) 121–174.Google Scholar
  13. [13]
    M. McQuillan. ‘Non-commutative Mori theory’ IHES pre-print March 2000, 100 pg.Google Scholar
  14. [14]
    M. McQuillan. ‘Comparisons of conformal structures via Macpherson’s graph construction’ in preparation. Google Scholar
  15. [15]
    M. McQuillan. ‘Deformation theory on inseparable scheme quotients’ in preparation. Google Scholar
  16. [16]
    Y. Miyaoka. ‘Deformation of morphisms along a foliation’ Proc. Symp. Math. 46 (1987).Google Scholar
  17. [17]
    S. Mori. ‘Threefolds whose canonical bundles are not numerically effective’ Ann. Math. 116 (1982) 133–176.zbMATHCrossRefGoogle Scholar
  18. [18]
    Y.-Y. Siu. Conversation at lunch, HKU 2000.Google Scholar
  19. [19]
    P. Vojta. ‘Diophantine approximation and value distribution theory’ Lect. Notes. in Math. 1239 (1997).Google Scholar
  20. [20]
    P. Vojta, P. ‘Siegel’s Theorem in the compact case’ Ann. Math. 133 (1991) 509–548.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    S.-T. Yau. ‘On the Ricci curvature of a complex Kahler manifold and the complex Monge-Ampère equation I’ Comm. Pure and App. Math. 31 (1978) 339–411.zbMATHCrossRefGoogle Scholar

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© Springer Basel AG 2001

Authors and Affiliations

  • Michael McQuillan
    • 1
  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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