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F. Campana :
Orbifolds and classification of algebraic varieties.
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We shall first introduce (with motivations) the category of (smooth) geometric orbifolds and the notion of special orbifold, then describe the birational structure of complex projective manifolds in terms of these notions (the LMMP provides the elementary steps of this description, but does not say how to iterate them). Finally, this description will be used to extend Lang's conjectures in arithmetics and hyperbolicity to arbitrary complex projective manifolds.
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J.-P. Demailly :
Jet differentials and hyperbolicity.
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The goal of the lectures will be to explain basic facts concerning the
use of jet differentials in the theory of hyperbolic varieties :
fundamental vanishing theorem for entire curves, Semple jet bundles
and invariant jet differentials. We will also hint the strategy used
to produce global jet differential, via the Riemann-Roch theorem or
via holomorphic Morse inequalities, and will illustrate these
techniques in the simpler case of generic surfaces of high degree in
the projective 3-space (proof of the Kobayashi conjecture).
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J. Duval :
Ahlfors currents and localization of entire curves.
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An Ahlfors current is a limit of a diverging
sequence of holomorphic discs. We'll see
how to construct entire curves inside the support
of such a current. As an application we'll give a
characterization of compact hyperbolic manifolds
in terms of an isoperimetric
inequality for holomorphic discs.
We'll also apply Ahlfors currents to recover some
classical results of value distribution theory.
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C. Gasbarri :
Nevanlinna theory and the ABC conjecture over function fields.
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We will begin by explaining the analogy between the arithmetic theory of algebraic points on a variety defined over a function field and the Nevanlinna theory of maps from finite coverings of the affine line on varieties. Then we will explain the main steps of the proof of the tautological inequality (in the Nevanlinna contest). Eventually, we will explain the inequalities of type $abc$ and the proof by McQuillan of the $1+\epsilon$ conjecture for curves.
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A. Levin :
Holomorphic curves and integral points.
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We will start with a brief review of the relevant concepts and the classical case of curves. We will then move on to a discussion of holomorphic curves and integral points on higher-dimensional varieties, discussing along the way the Nevanlinna-Diophantine dictionary of Vojta. Our primary focus will be on varieties with many components at infinity and on the methods originating in the work of Corvaja and Zannier (relying on Schmidt's Subspace Theorem/Cartan's Theorem).
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