Seminar Algebra and Geometry: Shengyuan Zhao (University of Rennes)

Birational Kleinian groups and birational structures

Let Y be a smooth complex projective surface. Let U be a connected Euclidean open set of Y. Let G be a subgroup of Bir(Y) which acts by holomorphic diffeomorphisms on U (i.e. preserves U and without indeterminacy points in U), in a free, properly discontinuous and cocompact way, so that the quotient X=U/G is a compact complex surface. Such a birational transformation group G, or more precisely such a quadruple (Y,U,G,X), will be called a birational Kleinian group. Once we have a birational Kleinian group, the quotient surface is equipped with a birational structure, i.e. an atlas of local charts with rational changes of coordinates. I will present some basic properties and subtleties of birational structures, compared to the classical geometric structures. Then I will begin by studying birational structures on a special type of non-algebraic surfaces, the Inoue surfaces, to reveal some of the general strategy. Using classification of solvable and abelian groups of the Cremona group, and by relating the foliations on Inoue surfaces with some birational dynamical systems via Ahlfors-Nevanlinna currents, I will show that the Inoue surfaces have one unique birational structure. Then I will move on to the general study of birational Kleinian groups with the additional hypothesis that the quotient surface is projective. I will explain how to use powerful results from Cremona groups, holomorphic foliations and non-abelian Hodge theory to get an almost complete classification of such birational Kleinian groups. 

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