Birational geometry is one of the current research trends in fields of Algebraic Geometry and Analytic Geometry. It came into prominence during mid-1980s, and has since seen a rise in research. The proposed lectures are on two topics of birational geometry.
Speaker : Yohan Brunebarbe
Title : Hyperbolicity and Fundamental groups.
Abstract : The course will focus on the interplay between the linear representations of the fundamental groups; the holomorphic (pluri) differentials and hyperbolicity of complex projective manifolds. It will also introduce the necessary tools and techniques of variational and mixed Hodge theories, to help prove that a local system of geometric origin on a 'special' manifold has a virtually abelian monodromy. The course will also explore on using non-abelian Hodge theory and harmonic maps to treat the general case.
Speaker : Frederic Campana
Title : Birational Geometry and Orbifold Pairs : Arithmetic and hyperbolic aspects.
Abstract : Birational geometry aims at deducing the qualitative structure of complex projective manifolds $X$ from the positivity/negativity properties of their canonical bundles $K_X$. The course will explain how to achieve this goal using the notion of 'special' manifolds, which generalise rational and elliptic curves. A manifold is indeed defined to be special if, for any p>0, no line subbundle of $\Omega^p_X$ has top Kodaira dimension $p$. The functorial 'core map' $c_X:X\to C_X$ then canonically splits any $X$ into its 'special' part (the fibres) and its `general type' part (the orbifold base). This spitting permits to give a simple (but still conjectural) description of the distribution of entire curves and rational points on any $X$.