Groups in Algebraic Geometry.
With the support of Forschergruppe 790
1 September 2008 - 15 November 2008Aims
Main topics will be : 1) Algebraic surfaces 2) Fundamental groups and uniformization problems (also ball and polydisk quotients) 3) Group actions, automorphisms, and birational automorphism groups 4) Galois groups, fields of definitions of algebraic varieties, Belyi type conjectures 5) Monodromy factorizations and symplectic topology of algebraic surfaces.
Purpose of the semester is a cross-fertilization of several expertises, many of them stemming from the study of algebraic surfaces, all centred on algebraic geometry, coming from diverse yet allied fields such as finite and infinite group theory, complex variables and uniformization, arithmetic, number theory and symplectic geometry.
Groups of transformations had an important role in algebraic geometry from the very onset, for instance Noether and Cremona studied the group of birational transformations of the plane, called the plane Cremona group. In spite of many classical results, even the classification of finite subgroups of the plane Cremona group was unresolved until very recently. Establishing the complete classification will be one important target. In another direction, groups of automorphisms of curves lead to very interesting questions about the socalled 'asymptotical group theory', in particular study of zeta functions of groups associated to the counting of subgroups of finite index. Curves with many automorphisms turn out to be rigid and defined over number fields, and the determination of their field of definition, in other words of their Grothendieck fundamental group, is a very important open problem. The theory of varieties isogenous to products of curves, and the so-called Beauville surfaces, leads to faithful actions of the absolute Galois group on components of moduli spaces, investigated through the change of fundamental groups for Galois conjugate varieties. An important target is to relate these and other geometrical descriptions with the arithmetic of uniformizing functions, especially of triangle coverings, when the inverse of the uniformizing function is related to the Gauss hypergeometric equation by classical work of H.A. Schwarz. The higher dimensional uniformization problems lead to a separation of the arithmetic and the nonarithmetic case, according to the investigations by Deligne and Mostow relating ball quotients to generalized Appell Lauricella hypergeometric integrals. Another important theme will be the still open Shafarevich's conjecture about the holomorphic convexity of the universal cover of an algebraic variety. We know from Koll'ar and Campana the existence of rational Shafarevich maps, but Bogomolov proposed one approach to disprove the conjecture in the case of fibred surfaces (relying on the study of Burnside type groups). Ball quotients relate on the one side to the study of certain very special moduli problems, on the other hand one would like to attack the classification of surfaces uniformized by the ball or the bidisk, and higher dimensional varieties of this type, using the expertise on arithmetic groups and asymptotic group theory. The study of biregular automorphisms leads also to the construction, initiated by Lucien Godeaux, of many interesting varieties as quotients by group actions, both free or with fixed points. The construction of examples is related, via Grauert and Remmert's generalized Riemann existence theorem, to the study of fundamental groups of open algebraic varieties and possible monodromies. Both are very active topics, the former is part of the fine classification of algebraic surfaces with small invariants (here a single quite explicit problem is the fine classification of numerical Godeaux surfaces). Surfaces with small invariants are extremely important not only for their intrinsic interest and beauty, but also for the purpose of establishing fine results for higher dimensional manifolds (e.g. : linear series, Noether type inequalities). The fine classification of surfaces of general type means the study of their moduli spaces, in particular of their irreducible and connected components, and of their structure. This problem, in view of the recent counterexamples by Manetti, Catanese and Wajnryb to the Def= Diff question, using mapping class group monodromies, ties in with the investigation of symplectic 4-manifolds, initiated by Donaldson, Gromov and carried out by Auroux and Katzarkov who defined symplectic invariants of branched coverings of the plane. This method appeals to the theory of braid monodromies, initiated by Chisini long ago to determine the fundamental groups of complements to cuspidal curves, and to the theory of corresponding horizontal and vertical braid monodromies developed by Moishezon and his school.