On real algebraic vector bundles

Supported by the Netherlands Organisation for Scientific Research (NWO)     

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 algebraic K-theory of real algebraic varieties with circle action

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S¹-action so that the quotient Y=X/S¹ is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with ,

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is onto, where , denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that

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for any real algebraic variety X. As an application we will show that for a compact connected Lie group G .     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 403–106, September, 1998.     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only.     

On meromorphic parameterizations of real algebraic curves

A singular flat geometry may be canonically assigned to a real algebraic curve ??; namely, via analytic continuation of unit speed parameterization of the real locus ${\Gamma_\mathbb{R}}$ . Globally, the metric ${\rho=|Q|=|q(z)|dzd\bar{z}}$ is given by the meromorphic quadratic differential Q on ?? induced by the standard complex form dx ²?+?dy ² on ${\mathbb{C}^2=\{(x,y)\}}$ . By considering basic properties of Q, we show that the condition for local arc length parameterization along ${\Gamma_\mathbb{R}}$ to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.     

On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

On sign determinations in real algebraic number fields

Summary Given an ordered fieldF and a finite ordered extensionE. If it is possible to perform constructively the four rational operations inF and to determine the sign for an arbitrary element ofF, a rule is given for the sign determination of an arbitrary element ofE.     

Coherent algebraic sheaves in real algebraic geometry

Summary In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

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Riassunto Nel presente lavoro si danno esempi di fibrati vettorialiF→X, definiti su varietà affini, regolari, compatte, connesseX che non ammettono struttura algebrica.

     

Non-embeddable real algebraic hypersurfaces

We study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems.     

On approximation of smooth submanifolds by nonsingular real algebraic subvarieties

Let M be a smooth submanifold of dimension m of a nonsingular real algebraic set X. If M can be approximated by nonsingular algebraic subsets of X, then the homology class in represented by M is algebraic. The converse, investigated in this paper, is true only in some exceptional cases.