These notes describe a general procedure for calculating the Betti numbers of the projective quotient varieties that geometric invariant theory associates to reductive group actions on nonsingular complex projective varieties. These quotient varieties are interesting in particular because of their relevance to moduli problems in algebraic geometry. The author describes two different approaches to the problem. One is purely algebraic, while the other uses the methods of symplectic geometry and Morse theory, and involves extending classical Morse theory to certain degenerate functions.

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Cohomology of Quotients in Symplectic and Algebraic Geometry

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Galois theory of linear differential equations

Some theorems on actions of algebraic groups

In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At the end of the paper, we discuss the recent counter-examples, in all dimensions greater than two, to the Markus-Yamabe conjecture (Global asymptotic Jacobian conjecture). Resume Dans ce papier nous presentons un rapport actualise sur les resultats les plus importants concernant la conjecture Jacobienne : plusieurs formulations equivalentes et diverses conjectures connexes sont considerees. A la fin du papier, nous donnons les contre-exemples recents, en toute dimension plus grande que deux, a la conjecture de Markus-Yamabe.

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Polynomial Automorphisms and the Jacobian Conjecture

Proof of the Deligne-Langlands conjecture for Hecke algebras

Some properties of the decompositions of algebraic varieties determined by actions of a torus

1. A recent note of D. J. Newman [3] shows that for polynomial maps of the real plane into itself injectivity implies surjectivity. The present note combines two independently obtained corrections and generalizations of Newman's proof. We need some preliminaries. Take the complex numbers as universal domain and define a real algebraic set to be the set of real points of an algebraic set that is defined over the reals. A real algebraic set V is Zariski-dense in its Zariski closure V, which is an algebraic set defined over the reals, and V is the set of real points of V. Any algebraic subset of V meets V in a real algebraic subset. V is irreducible (in its Zariski topology), i.e. V is a real algebraic variety, if and only if V is irreducible (over the complex numbers), we define dim V =dim V, and we call PC V simple if P is a simple point of V; such points P exist, and there exist uniformizing parameters that are defined over the reals for 7 at such a point P, hence real local power series expansions, so that at each of its simple points V, in its ordinary topology, is locally a real analytic manifold of dimension dim V. For real algebraic sets V, W define a rational map f: V-*W to be the restriction to VX W of a rational map f: V--W that is defined over the reals; the rational map f: V-*W is a morphism if f is defined at each point of V. Supposing the morphism of real algebraic varieties f: V-*W to be such that f(V) is Zariski-dense in W, a simple point PE V may be found such that f(P) is simple on Wand df has the correct rank dim V-dim W at P, implying that, for the real analytic structure of V, W at P, f(P) respectively, f is locally a projection onto a direct factor; in particular, if f is finite-to-one, then dim V=dim W

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Andrzej Białynicki-Birula

Papers

Some theorems on actions of algebraic groups

Some properties of the decompositions of algebraic varieties determined by actions of a torus

Injective morphisms of real algebraic varieties

On the Representation of Quasi-Boolean Algebras

Remarks on the action of an algebraic torus on kn. II