K-theory of real algebraic surfaces and threefolds

1989 ◽  
Vol 106 (3) ◽  
pp. 471-480 ◽  
Author(s):  
J. Bochnak ◽  
W. Kucharz
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Algebraic Surfaces ◽  
Real Algebraic Variety ◽  
Projective Modules ◽  
Algebraic Subset ◽  
Witt Group ◽  
Regular Functions ◽  
Real Algebraic Surfaces ◽  
Continuous Complex

LetXbe an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Letdenote the ring of regular functions onX([6], chapter 3). (IfXis an algebraic subset of ℝnthenis comprised of all functions of the formf/g, whereg, f: X→ ℝ are polynomial functions withg−1(O) = Ø.) In this paper, assuming thatXis compact, non-singular, and that dimX≤ 3, we compute the Grothendieck groupof projective modules over(cf. Section 1), and the Grothendieck groupand the Witt groupof symplectic spaces over(cf. Section 2), in terms of the algebraic cohomology groupsandgenerated by the cohomology classes associated with the algebraic subvarieties ofX. We also relate the groupto the Grothendieck groupKO(X) of continuous real vector bundles overX, and the groupsandto the Grothendieck groupK(X)of continuous complex vector bundles overX.

scholarly journals Symplectic bilinear forms on affine real algebraic surfaces

1989 ◽  
Vol 31 (2) ◽  
pp. 195-198
Author(s):  
W. Kucharz
Keyword(s):  
Commutative Ring ◽  
Algebraic Surfaces ◽  
Bilinear Forms ◽  
Witt Group ◽  
Real Algebraic Surfaces ◽  
Definition Of

Given a commutative ring A with identity, let W–1(A) denote the Witt group of skew-symmetric bilinear forms over A (cf. [1] or [7] for the definition of W–1 (A)).

scholarly journals Symplectic complex bundles over real algebraic four-folds

1989 ◽  
Vol 47 (3) ◽  
pp. 430-437
Author(s):  
Wojciech Kucharz
Keyword(s):  
Algebraic Variety ◽  
Real Algebraic Variety ◽  
Bilinear Forms ◽  
Cohomology Groups ◽  
Witt Group ◽  
Regular Functions

AbstractLetXbe a compact affine real algebraic variety of dimension 4. We compute the Witt group of symplectic bilinear forms over the ring of regular functions fromXto C. The Witt group is expressed in terms of some subgroups of the cohomology groups.

scholarly journals Complex vector bundles on real algebraic varieties of small dimension

1988 ◽  
Vol 38 (3) ◽  
pp. 345-349
Author(s):  
Wojciech Kucharz
Keyword(s):  
Algebraic Variety ◽  
Vector Bundles ◽  
Small Dimension ◽  
Real Algebraic Variety ◽  
Algebraic Varieties ◽  
Complex Vector ◽  
Real Algebraic Varieties ◽  
Continuous Complex

LetXbe an affine real algebraic variety. In this paper, assuming that dimX≤ 7 and thatXsatisfies some other reasonable conditions, we give a characterisation of those continuous complex vector bundles onXwhich are topologically isomorphic to algebraic complex vector bundles onX.

scholarly journals On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces

2000 ◽  
Vol 68 (1) ◽  
pp. 41-54
Author(s):  
Edoardo Ballico
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Algebraic Surfaces ◽  
Stable Rank ◽  
Projective Surface ◽  
Chern Classes ◽  
The Real ◽  
Smooth Projective Surface ◽  
Real Algebraic Surfaces

AbstractLet X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 365-427 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Positive Cone ◽  
Point Of View ◽  
Order Structure ◽  
Topological Spaces ◽  
Projective Modules ◽  
Finitely Generated ◽  
C Algebras ◽  
Isomorphism Classes

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

scholarly journals A remark on projective modules

1987 ◽  
Vol 36 (3) ◽  
pp. 515-520
Author(s):  
Wojciech Kucharz
Keyword(s):  
Projective Modules ◽  
Real Numbers ◽  
Algebraic Subset ◽  
Regular Functions ◽  
The Ideal

Let R denote the field of real numbers and let A be the ring of regular functions on Rn, that is the localization of R[T1 …, Tn] with respect to the set of all polynomials vanishing nowhere on Rn. Let X be an algebraic subset of Rn and let I(X) be the ideal of A of all functions vanishing on X. Assume that X is compact and nonsingular and k = codim X = 1, 2, 4 or 8. we prove here that if the A/I(X)-module I(X)/I(X)2 can be generated by k elements, then there exist a projective A-module P of rank k and a homomorphism from P onto I(X).

scholarly journals Geometrically rational real conic bundles and very transitive actions

2010 ◽  
Vol 147 (1) ◽  
pp. 161-187 ◽  
Author(s):  
Jérémy Blanc ◽  
Frédéric Mangolte
Keyword(s):  
Automorphism Groups ◽  
Algebraic Surfaces ◽  
Algebraic Models ◽  
Transitive Automorphism ◽  
Real Locus ◽  
Real Algebraic Surfaces ◽  
Conic Bundles ◽  
Transitive Actions ◽  
Insight Into

AbstractIn this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

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