Andrzej Białynicki-Birula

INJECTIVE MORPHISMS OF REAL ALGEBRAIC VARIETIES

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 Newmans 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

Finiteness of the number of maximal open subsets with good quotients

The aim of the paper is to prove the Main Theorem which says that, in any complex normal variety with an action of a reductive groupG, there are only finitely many subsets which are maximal in the family of all openG-invariant subsets admitting a good quotient.

Quotients by Actions of Groups

The aim of this survey is to present the main trends and directions of research in the theory of quotients by group actions. The theory, in the assumed here sense, contains Geometric Invariant Theory understood as a theory providing geometric interpretations of rings and more generally sheaves of invariant functions determined by group actions. Though the affine (i.e. local) case plays the basic role in the theory, we shall put attention mainly on results of global character. The affine case should be considered rather as a part of Classical Invariant Theory, because translation of algebraic theory of rings of invariants into the geometric language is usually a matter of routine.

On the inverse problem of Galois theory of differential fields

1. All fields considered here are of characteristic 0. Let F be a field, let C be an algebraically closed subfield of F. Let G be a connected algebraic group defined over C. F(G) denotes the field of all rational functions on G defined over F. If gCG then F(g) denotes the field generated by g over F. We shall say that a derivation of F(G) commutes with G*(C) if it commutes with g*, for every gEG(C), where g* denotes the automorphism of F(G) induced by the left translation by g, i.e., (g*f)(x) =f(gx), for any xCG. F denotes the Lie algebra of all derivations of F(G) that are zero on F and which commute with G*(F). If G1 is a normal subgroup of G defined over F then F(G/G1) is canonically isomorphic to a subfield of F(G); we shall identify F(G/G1) and this subfield. If R is an integral domain then (R) denotes the field of fractions of R. Every derivation d of R can be uniquely extended to a derivation of R (the extended derivation will be also denoted by d). If F1, F2 are two fields containing F as a subfield and if d1, d2 are derivations of F1, F2, respectively, such that d1i F= d2 I F and d1(F) C F then d1i d2 denotes the derivation of F1,0F F2 determined by (d1 d2)(a 0 b) -d1(a)Gb+a0d2(b), for every aGF1 and bCF2. do denotes the zero derivation of a field (it will be always clear what field we have in mind). The underlying field of an ordinary differential field 53 will be denoted by F.