Wojciech Kucharz

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.

Rational maps in real algebraic geometry

The paper deals with rational maps between real algebraic sets. We are interested in the rational maps which extend to continuous maps defined on the entire source space. In particular, we prove that every continuous map between unit spheres is homotopic to a rational map of such a type. We also establish connections with algebraic cycles and vector bundles.

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).

Stratified-algebraic vector bundles

Abstract We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle ξ on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification 𝒮 {\mathcal{S}} of X such that the restriction of ξ to each stratum S in 𝒮 {\mathcal{S}} is an algebraic vector bundle on S. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles.

ON RELATIVE REAL HOLOMORPHY RINGS

LetF be a formally real function field (i.e., −1 is not a sum of squares inF) of transcendence degreen over ℝ. LetA be a finitely generated ℝ-subalgebra ofF whose field of quotients isF. The real holomorphy ringH(F|A) ofF overA is the intersection of all valuation rings ofF which containA and have formally real residue field. The ringH(F|A) has been extensively studied and applied by Becker, Schülting and others. It is known to be a Prüfer domain of Krull dimension not exceedingn and from this it can be shown that every finitely generated ideal ofH(F|A) can be generated byn+1 elements. Here, assuming thatA is regular (i.e., every localization with respect to a prime ideal is regular local), we give necessary and sufficient topological conditions in order for every finitely generated ideal ofH(F|A) to admitn generators. We also provide a geometric description ofH(F|A).

Curve-rational functions

Let W be a subset of the set of real points of a real algebraic variety X. We investigate which functions