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Fano foliations 0 - Algebraicity of smooth formal schemes and applications to foliations - lecture 1

By Stéphane Druel

Also appears in collection : Jean-Morlet Chair 2020 - Research School: Geometry and Dynamics of Foliations / Chaire Jean-Morlet 2020 - Ecole : Géométrie et dynamiques des feuilletages

In the last few decades, much progress has been made in birational algebraic geometry. The general viewpoint is that complex projective manifolds should be classified according to the behavior of their canonical class. As a result of the minimal model program (MMP), every complex projective manifold can be built up from 3 classes of (possibly singular) projective varieties, namely, varieties $X$ for which $K_X$ satisfies $K_X<0$, $K_X\equiv 0$ or $K_X>0$. Projective manifolds $X$ whose anti-canonical class $-K_X$ is ample are called Fano manifolds.

Techniques from the MMP have been successfully applied to the study of global properties of holomorphic foliations. This led, for instance, to Brunella's birational classification of foliations on surfaces, in which the canonical class of the foliation plays a key role. In recent years, much progress has been made in higher dimensions. In particular, there is a well developed theory of Fano foliations, i.e., holomorphic foliations $F$ on complex projective varieties with ample anti-canonical class $-K_F$.

This mini-course is devoted to reviewing some aspects of the theory of Fano Foliations, with a special emphasis on Fano foliations of large index. We start by proving a fundamental algebraicity property of Fano foliations, as an application of Bost's criterion of algebraicity for formal schemes. We then introduce and explore the concept of log leaves. These tools are then put together to address the problem of classifying Fano foliations of large index.

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Citation data

  • DOI 10.24350/CIRM.V.19630603
  • Cite this video Druel, Stéphane (30/04/2020). Fano foliations 0 - Algebraicity of smooth formal schemes and applications to foliations - lecture 1. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19630603
  • URL https://dx.doi.org/10.24350/CIRM.V.19630603

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Bibliography

  • Araujo Carolina, Druel Stéphane: Characterization of generic projective space bundles and algebraicity of foliations. Comment. Math. Helv. 94 (2019), 833-853 - http://dx.doi.org/10.4171/CMH/475
  • ARAUJO, Carolina et DRUEL, Stéphane. On Fano foliations 2. In : Foliation Theory in Algebraic Geometry. Springer, Cham, 2016. p. 1-20. - https://doi.org/10.1007/978-3-319-24460-0_1
  • ARAUJO, Carolina et DRUEL, Stéphane. On fano foliations. Advances in Mathematics, 2013, vol. 238, p. 70-118. - https://doi.org/10.1016/j.aim.2013.02.003
  • CAMPANA, Frédéric et PĂUN, Mihai. Foliations with positive slopes and birational stability of orbifold cotangent bundles. Publications mathématiques de l'IHÉS, 2019, vol. 129, no 1, p. 1-49. - https://doi.org/10.1007/s10240-019-00105-w
  • BOGOMOLOV, Fedor et MCQUILLAN, Michael. Rational curves on foliated varieties. In : Foliation theory in algebraic geometry. Springer, Cham, 2016. p. 21-51. - http://dx.doi.org/10.1007/978-3-319-24460-0_2
  • Bost, Jean-Benoît. Algebraic leaves of algebraic foliations over number fields. Publications Mathématiques de l'IHÉS, Tome 93 (2001) , pp. 161-221 - http://www.numdam.org/item/PMIHES_2001__93__161_0/
  • Bost, J.-B.; Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems ii. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N., Loeser, F. (eds.) Geometric Aspects of Dwork Theory, vol. I. Walter de Gruyter II, Berlin (2004). -

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