J. Bochnak

Real algebraic geometry

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilberts 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

Algebraic models of smooth manifolds

In fact, we conjecture that such a family can be chosen uncountable. In view of this theorem it seems natural and interesting to investigate algebrogeometric properties of various algebraic models of a given smooth manifold. Let us now describe one of these problems, which will be the main object of study in this paper. For notions, definitions and results of real algebraic geometry we refer the reader to the book [9]. Given a compact nonsingular real affine algebraic variety X, denote by H~Ig(x , 7Z/2) the subgroup of nk(X, Z/2) of homology classes represented by (Zariski closed) k-dimensional algebraic subvarieties of X (cf. [9] Chap. 1 1). Let H~lg(X, Z/2) be the image of H~a~k(X, 71/2), d=dimX, under the Poincar6 duality isomorphism Hd_k(X, 71/2)-o Hk(X, 71/2). These groups H~Ig(X, 71/2) are undoubtedly one of the most important invariants of an algebraic variety X. Still, our knowledge of the behavior of H*~g(X, 71/2) is rather limited. The main open problem is the following.

K -theory of real algebraic surfaces and threefolds

Let X be an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝ n . (For definitions and notions of real algebraic geometry we refer the reader to the book [ 6 ].) Let denote the ring of regular functions on X ([ 6 ], chapter 3). (If X is an algebraic subset of ℝ n then is comprised of all functions of the form f/g , where g, f: X → ℝ are polynomial functions with g −1 (O) = O.) In this paper, assuming that X is compact, non-singular, and that dim X ≤ 3, we compute the Grothendieck group of projective modules over (cf. Section 1), and the Grothendieck group and the Witt group of symplectic spaces over (cf. Section 2), in terms of the algebraic cohomology groups and generated by the cohomology classes associated with the algebraic subvarieties of X . We also relate the group to the Grothendieck group KO(X ) of continuous real vector bundles over X , and the groups and to the Grothendieck group K(X) of continuous complex vector bundles over X .

On homology classes represented by real algebraic varieties

Introduction. Let X be a compact real algebraic variety (essentially, a compact real algebraic set, see Section 1). Denote by H k (X,Z/2) the subgroup of Hk(X,Z/2) generated by the homology classes represented by Zariski closed k-dimensional subvarieties of X. If X is nonsingular and d = dimX, let H alg(X,Z/2) be the subgroup of H(X,Z/2) consisting of all the cohomology classes that are sent via the Poincare duality isomorphism H(X,Z/2)→ Hd−k(X,Z/2) into H d−k(X,Z/2). In this short paper we survey certain results concerning the groups H k and H k alg, and their applications. Section 1 contains the precise definitions of these groups (based on a construction of the fundamental homology class of a compact real algebraic variety) and theorems establishing their functorial properties and relating them to the StiefelWhitney classes of algebraic vector bundles. With the exception of Theorem 1.7 (ii), all the results (modulo minor modifications) come from the classical source [15]. In Section 2 we adopt a scheme-theoretic point of view and discuss how the groups H k are related to the theory of algebraic cycles, especially rational and algebraic equivalence. In particular, we give a purely algebraic geometric description of H k . Our main references are [15, 17, 25, 28]. In Section 3 we study how the groups H alg(X,Z/2), for k = 1, 2, vary as X runs through the class of nonsingular real algebraic varieties diffeomorphic to a given closed C∞ manifold. We rely mostly on [9, 11, 37], but the reader may also wish to consult

Morphisms, line bundles and moduli spaces in real algebraic geometry

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ALGEBRAIC CYCLES AND APPROXIMATION THEOREMS IN REAL ALGEBRAIC GEOMETRY

Let Af be a compact C°° manifold. A theorem of Nash-Tognoli asserts that M has an algebraic model, that is, M is diffeomorphic to a nonsingular real algebraic set X. Let FV^AfX, Z/2) denote the subgroup of Hk(X, Z/2) of the cohomology classes determined by algebraic cycles of codi- mension k on X. Assuming that M is connected, orientable and dim M > 5 , we prove in this paper that a subgroup G of H2(M, Z/2) is isomorphic to H^ (X, Z/2) for some algebraic model X of M if and only if w2(TM) is in G and each element of G is of the form W2K) for some real vector bundle £ over M , where w2 stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups G of HX(M, Z/2).

Topology of Real Algebraic Varieties

In the first section, we prove some combinatorial topological properties of real algebraic sets; the simplest and most important of these properties is the fact that, for every semi-algebraic triangulation of a bounded algebraic set of dimension d and every (d - 1)-simplex σ of such a triangulation, the number of d-simplices of the triangulation having σ as a face is even. In the second section, we use this property and an appropriate stratification to prove that, for every point a of an algebraic set V, the local Euler-Poincare characteristic χ(V, V \ a) is odd; this result gives a necessary combinatorial condition for a polyhedron to be homeomorphic to a real algebraic set. In Section 3 we define the fundamental ℤ/2-homology class of a real algebraic variety. This leads to the concept of algebraic homology groups of a real algebraic variety, consisting of the homology classes represented by algebraic subsets. These groups, which are basic invariants, will be used in Chap. 12 and 13. We construct examples of nonsingular algebraic sets whose homology is not totally algebraic. In Section 4, we use the Borel-Moore fundamental classes to prove that an injective regular mapping from a nonsingular irreducible algebraic set to itself is surjective. The analogous result in complex algebraic geometry (without the assumption of nonsingularity) is well known, but the methods of proof are completely different. Section 5 contains an upper bound for the sum of the Betti numbers of an algebraic set. Section 6 is devoted to algebraic curves in the real projective plane. We prove Harnack’s theorem concerning the maximum number of connected components of a nonsingular curve of given degree and some results concerning the first part of Hilbert’s 16th problem (without proving the crucial Rokhlin congruence, Theorem 11.6.4).

Sur le theoreme des zeros de Hilbert “differentiable”

Ce probleme est un cas particulier du probleme general suivant: Soient X un espace topologique (ou un germe d’espace topologique) et A(X) un anneau de fonctions reelles ou complexes sur X. Pour un ideal J de A(X) notons traditionnellement, lot J = {x E X : q(x) = 0 pour tout cp E J} l’ensemble des zeros de J, et id lot J = {q E A(X) : cpI lot J = 0) l’ideal de fonctions s’annu lent sur lot J. En supposant que l’anneau A(X) est raisonnable on peut poser le:

Smooth maps and real algebraic morphisms

Let X be a compact nonsingular real algebraic variety and letY be either the blowup of Pn(R) along a linear subspace or a nonsingular hypersurface of Pm(R) × Pn(R) of bidegree (1, 1). It is proved that a C map f : X → Y can be approximated by regular maps if and only if f ∗ ( H1(Y, Z/2) ) ⊆ H1 alg (X, Z/2), where H1 alg (X, Z/2) is the subgroup of H1(X, Z/2) generated by the cohomology classes of algebraic hypersurfaces in X. This follows from another result on maps into generalized flag varieties. Received by the editors September 24, 1997; revised October 16, 1998. Both authors were partially supported by NATO Collaborative Research Grants Programme, CRG 960011. The second author was partially supported by NSF Grant DMS-9503138. AMS subject classification: 14P05, 14P25. c ©Canadian Mathematical Society 1999. 445

ON DIVIDING REAL ALGEBRAIC VARIETIES

Every nonsingular projective real algebraic curve C has a unique, up to isomorphism over ℝ, nonsingular projective complexification V . If V \ C is disconnected, then C is said to be dividing (we identify C with the set of real points of V ). Classical results supply numerous examples of dividing real curves. This class of curves was first studied by Felix Klein [ 11 ]. A characterization of dividing real curves is given in [ 6 ]. In higher dimensions the situation is more complicated. First of all, every nonsingular projective real algebraic variety X of dimension d always has several nonisomorphic nonsingular projective complexifications, provided that d [ges ]2. Furthermore, if d [ges ]2 and W is a nonsingular projective complexification of X , then W \ X is connected (we identify X with the set of real points of W ). What does it then mean for X to be dividing? An answer can be given in terms of homology theory. Let K be a principal ideal domain. Assume that X , regarded as a topological manifold, is orientable over K . We say that X is dividing over K if there exists a fundamental homology class of X over K whose image in H d ( W , K ) under the homomorphism induced by the inclusion map X [rarrhk ] W is zero. In the present paper we prove that this definition does not depend on the choice of W , and give a characterization of real varieties dividing over ℚ. For more information on real varieties dividing over ℤ/2 the reader may consult [ 18 ] (please note that terminology in [ 18 ] is different than here). We also discuss the relationship between real varieties dividing over ℚ (or ℤ) and dividing over ℤ/2. It follows from well-known facts that a real curve is dividing over K if and only if it is dividing.