On the picard group and the brauer group of a real algebraic surface

The map of the Brauer group of a real algebraic surface to the invariant part of the Brauer group of its complexification is studied. In this study, the real cycle map of the Picard group is used. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 211–220, February, 2000.     

The Brauer group of a noncomplete real algebraic surface

The Brauer group of a noncomplete real algebraic surface is calculated. The calculations make use of equivariant cohomology. The resulting formula is similar to the formula for a complete surface, but the proof is substantially different. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 355–359, March, 2000.     

On the fundamental homology classes of a real algebraic variety

It is proved that there is only one relation between the homology classes determined by the real points of a special real algebraic variety. This relation is equal to the sum of all the homology classes. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 216–219, August, 1999.     

The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E₂-term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G-fibration ƒ: Y → X, where G is a finite group. We show that if the base-G-spaceX is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X/G.     

Albanese homomorphism of the Chow group of 0-cycles of a real Algebraic variety

We study a certain homomorphism of the Chow group of 0-cycles of degree zero of a real algebraic variety into the group of real points of the Albanese variety; this homomorphism is obtained from the Albanese mapping for the corresponding variety. The kernel of this homomorphism is calculated and estimates for the kernel of the mapping of the torsion groups are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 76–83, January, 1999.     

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53-64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119-171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173-177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91-117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q-forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17-22, 2001, TIFR, Mumbai, 2001, pp. 469-490] of D. Morris (Witte) on a completely different topic. We will answer that question too.     

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.     

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex. This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)     

Recent researches on multivariate spline and piecewise algebraic variety

The multivariate splines as piecewise polynomials have become useful tools for dealing with Computational Geometry, Computer Graphics, Computer Aided Geometrical Design and Image Processing. It is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials. Recently the surfaces are mainly represented by multivariate piecewise polynomials (i.e. multivariate splines), so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry. Moreover, the piecewise algebraic variety will be also important in computational geometry, computer graphics, computer aided geometrical design and image processing. The purpose of this paper is to introduce some recent researches on multivariate spline, piecewise algebraic variety (curve), and their applications.     

Real Algebraically Maximal Varieties

For real algebraic varieties whose real algebraic cohomology group is maximal, a canonical homomorphism is constructed from the cohomology group of the set of complex points into the cohomology group of the set of real points, and then it is proved that this homomorphism is an isomorphism.     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

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.     

Twisted holomorphic forms on generalized flag varieties

In this paper we prove some vanishing theorems for the twisted Dolbeault cohomology of the complete flag varieties associated to a simple, simply connected algebraic group.     

Homogeneous varieties under split solvable algebraic groups

We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.     

Two-primary algebraic -theory of rings of integers in number fields

We relate the algebraic -theory of the ring of integers in a number field to its étale cohomology. We also relate it to the zeta-function of when is totally real and Abelian. This establishes the -primary part of the ``Lichtenbaum conjectures.' To do this we compute the -primary -groups of and of its ring of integers, using recent results of Voevodsky and the Bloch-Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory.

     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

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.

     

The BGQ spectral sequence for noncommutative spaces

We prove an analogue of the Brown-Gersten-Quillen (BGQ) spectral sequence for noncommutative spaces. As applications, we consider this spectral sequence over affine and projective spaces associated to right fully bounded noetherian (FBN) rings.