On homology of real algebraic varieties

Let be a commutative ring with unity and an -oriented compact nonsingular real algebraic variety of dimension . If is any nonsingular complexification of , then the kernel, which we will denote by , of the induced homomorphism is independent of the complexification. In this work, we study and give some of its applications.

     

On the algebraic K-theory of the complex K-theory spectrum

Let p≥5 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. In this paper we study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v ₁) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over the polynomial algebra \mathbbF_p[b]{\mathbb{F}}_{p}[b], where b is a class of degree 2p+2 defined as a “higher Bott element”.     

On products in algebraic K-theory

This paper investigates the product structure in algebraic K-theory of rings. The first objective is to understand the relationships between products and the kernel of the Hurewicz homomorphism relating the algebraic K-theory of any ring to the integral homology of its linear groups. The second part of the paper is devoted to the ring of integers . Using recent results of V. Voevodsky we completely determine the products in tensored with the ring of 2-adic integers. Received: January 3, 1999.     

Submanifolds of real algebraic varieties

By the Nash-Tognoli theorem, each compact smooth manifold is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of . We construct algebraic models of with controlled behavior of the group of cohomology classes represented by algebraic subsets of .

     

Vector bundles over real algebraic varieties

The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672.     

Virtual Betti numbers of real algebraic varieties

We show that for all i?0 the i-th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti numberβ_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X then β_i(X)=β_i(X?Y)+β_i(Y). We show by example that there is no natural weight filtration on the $\text{Z}{}_2$-cohomology of real algebraic varieties with compact supports such that the virtual Betti numbers are the weighted Euler characteristics. To cite this article: C. McCrory, A. Parusiński, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The topology of two-dimensional real algebraic varieties

Sunto È noto che ogni spazio analitico reale è localmente omeomorfo al cono su un poliedro con caratteristica di Eulero-Poincaré pari. Si dimostra che questa condizione è anche sufficiente affinchè un poliedro (compatto) di dimensione due P sia omeomorfo ad una varietà algebrica reale affine P. Segue inoltre dalla costruzione che la P ottenuta ha, in un certo senso, un insieme di singolarità algebriche minimale, compatibilmente con la topologia di P.

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The authors are members of the G.N.S.A.G.A.     

Estimates of characteristic numbers of real algebraic varieties

We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than d, and for the size of the sum of Betti numbers with Z/2 coefficients for the real form of complex manifolds of complex degree less than d.     

Positive polynomials on projective limits of real algebraic varieties

We reveal some important geometric aspects related to non-convex optimization of sparse polynomials. The main result, a Positivstellensatz on the fibre product of real algebraic affine varieties, is iterated to a comprehensive class of projective limits of such varieties. This framework includes as necessary ingredients recent works on the multivariate moment problem, disintegration and projective limits of probability measures and basic techniques of the theory of locally convex vector spaces. A variety of applications illustrate the versatility of this novel geometric approach to polynomial optimization.     

On real algebraic vector bundles

Supported by the Netherlands Organisation for Scientific Research (NWO)     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

The Leibniz formula in algebraic K-theory

The paper may be viewed as an addendum to a paper of Thomason and Throbaugh, where the K-theory of algebraic varieties is equipped with relative K-groups. It is proved that this enriched K-theory satisfies the Panin—Smirnov axioms for ring cohomology theories of algebraic varieties. In particular, it is proved that the Leibniz formula, describing the interaction between multiplication and differential, holds in this case. The language of symmetric spectra and of monoidal model categories is used. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 264–292.