Constructive Hyperbolicity

speaker: Mikhail Zaidenberg (Universite de Grenoble I - Institut Fourier)

abstract: Around 1970, S. Kobayashi suggested that a general hypersurface $D$ of high enough degree ($\ge 2n+1$) in ${\bf P}^n$ is hyperbolic (i.e., every holomorphic map ${\bf C}\to D$ is constant) and has hyperbolic complement. Since then, a significant progress was done in a work by Green and Griffiths, Siu and Yeung, Demailly and El Goul, McQuillan and Brunella, e.a. After giving a brief survey on this development, we will mainly concentrate on the methods which allow to obtain constructive examples of hyperbolic hypersurfaces and their complements.

An approximate content of the lectures is the following one: 1. Kobayashi hyperbolicity and Brody entire curves. 2. Absorbing stratifications and stability of hyperbolicity. 3. Algebraic hyperbolicity of projective hypersurfaces and their complements. 4. Smooth quintics in ${\bf P}^2$ with hyperbolic complements. 5. M. Green's value distribution theorems. Hyperbolic hypersurfaces in ${\bf P}^n$ of Fermat-Waring type. 6. Hyperbolic hypersurfaces in ${\bf P}^2$ and ${\bf P}^3$ provided by symmetric products of curves. 7. Hyperbolic non-percolation. Examples of degree 8 hyperbolic surfaces in ${\bf P}^3$. 8. Perspectives.

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