Asymptotic behaviour of Betti numbers of real algebraic surfaces

Let be a nonsingular real algebraic surface of degree m in the complex projective space and its real point set in . In the spirit of the sixteenth Hilbert's problem, one can ask for each degree m about the maximal possible value of the Betti number (i=0 or 1). We show that is asymptotically equivalent to for some real number and prove inequalities and . Received: April 26, 2000     

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.     

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 .

     

Virtual Betti numbers of compact locally symmetric spaces

We show that the virtual Betti number of a compact locally symmetric space with arithmetic fundamental group is either the Betti number of the compact dual or else is infinite.     

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.

     

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.     

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 .     

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.     

Representation varieties of arithmetic groups and polynomial periodicity of Betti numbers

We give a method of studying character varieties of arithmetic groups with an application to polynomial periodicity of Betti numbers of manifolds in congruence towers.     

On approximation of real numbers by algebraic numbers of bounded degree

Dirichlet proved that for any real irrational number ξ there exist infinitely many rational numbers p/q such that |ξ−p/q|<q⁻². The correct generalization to the case of approximation by algebraic numbers of degree ?n, n>2, is still unknown. Here we prove a result which improves all previous estimates concerning this problem for n>2.     

Picard and Lefschetz numbers of real algebraic surfaces

Two Picard numbers and two Lefschetz numbers are defined for a real algebraic surface. They are similar to the Picard number and the Lefschetz number of a complex algebraic surface. For these numbers, some estimates and relations in the form of inequalities are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 847–852, June, 1998.     

Maximal Betti numbers

We provide a short proof that the lexicographic ideal has the greatest Betti numbers among all graded ideals with a fixed Hilbert function.

     

On Chern numbers of algebraic varieties with arbitrary singularities

In 1965 the author introduced the notion of Chern classes for an algebraic variety with arbitrary singularities. Based on this definition the well-known Miyaoka-Yao inequalities have been proved and extended by quite simple direct computations.     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$ . In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.