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M. Brion Institut Fourier Grenoble, France Michel.Brion@ujf-grenoble.fr

Anti-affine algebraic groups
In this talk, we scheme G of finite smooth, connected also their universal varieties. introduce and study the class of groups of the title. We say type over a field k is anti-affine, if O(G) = k . Then G is and commutative. Examples include, of course, all abelian vector extensions (in characteristic zero only) and certain that a group known to be varieties, but semi-abelian

The class of anti-affine groups and the class of affine (or, equivalently, linear) group schemes play complementary roles in the structure of group schemes over fields. Indeed, any connected group scheme G, of finite type over k , has a largest anti-affine subgroup scheme Gant . This subgroup is contained in the centre of G, and the quotient G/Gant is the affinization of G, i.e., the universal affine quotient group scheme. Also, G has a medest normal connected affine subgroup scheme Gaff such that G/Gaff is an abelian variety. This yields the Rosenlicht decomposition : G = Gaff Gant and Gaff Gant contains (Gant )aff ; moreover, the quotient group scheme (Gaff Gant )/(Gant )aff is finite. Affine group schemes have been extensively investigated, but little seems to be known about their anti-affine counterparts; they only appear implicitly in work of Rosenlicht and Serre. Here we present some fundamental properties of anti-affine groups, which reduce their structure to that of abelian varieties. Our main result classifies anti-affine algebraic groups G over an arbitrary field. In positive characteristics, G is a semi-abelian variety, parametrized by a pair (A, ) where A is an abelian variety and is a sublattice of the group of geometric points of A, stable unde the absolute Galois group. The classification is a bit more complicated in characteristic zero: the parameters are then triples (A, , V ) where A and are as above, and V is a subspace of the Lie algebra of A. In both cases, A is the dual of the abelian variety G/Gaff . As a consequence, every anti-affine group over a finite field is an abelian variety. Combined with the Rosenlicht decomposition, it follows that any connected group scheme G over a finite field has a decomposition G = Gaff Gab , where Gab is the largest abelian subvariety of G; moreover, Gaff Gab is finite. For algebraic groups, this result is due to Arima. Our classification also implies a structure result for connected algebraic groups G over any perfect field k of positive characteristic, namely, the decomposition G = Guni S where Guni Gaff denotes the medest normal connected subgroup such that Gaff /Guni is a torus, and S G is a semi-abelian subvariety; moreover, Guni S is finite. If k is algebraically closed, then the group Guni is generated by all connected unipotent subgroups of G. Another application concerns Hilbert's fourteenth problem in its algebro-geometric formulation: does any quasi-affine variety have a finitely generated coordinate ring? The answer is known to be negative, the first counterexample being due to Rees. Here we obtain many counterexamples, namely, all Gm -torsors associated to ample line bundles over anti-affine, non-complete groups.

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