V. A. Krasnov

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.

Equivariant topological classification of the Fano varieties of real four-dimensional cubics

The equivariant topological type of the Fano variety parametrizing the set of lines on a nonsingular real hypersurface of degree three in a five-dimensional projective space is calculated. In the investigation of this Fano variety, results and constructions of the paper by Finashin and Kharlamov on the rigid projective classification of real four-dimensional cubics are used. The construction of Hassett (from the paper devoted to special four-dimensional cubics) is also applied.

On the Fano variety of a class of real four-dimensional cubics

The topological type of the real part of the Fano variety parametrizing the set of lines on a nonsingular real hypersurface of degree three in a five-dimensional projective space is evaluated provided that the hypersurface belongs to a special rigid projective class. In the paper by Finashin and Kharlamov on the rigid projective classification of real four-dimensional cubics, this class is said to be irregular. The results of the author of the present paper from the article devoted to the equivariant topological classification of the Fano varieties of real cubic fourfolds are also used.

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.

Real four-dimensional GM-triquadrics

Nonsingular maximal intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensionalM-triquadrics. The dimensions of their cohomology spaces with coefficients in the field of two elements are calculated.

Real GM-biquadrics

We consider real algebraic varieties that are the intersection of two real quadrics. For brevity, we refer to such varieties as real biquadrics. The rigid isotopy classes of real biquadrics have been described long ago. In the present paper, we find the rigid isotopy classes in which the biquadrics are GM-varieties.

Real two-dimensional intersections of a quadric by a cubic

The paper is devoted to finding the rigid isotopy classes of real projective surfaces that are obtained from nonsingular cubic sections of a chosen nonsingular real quadric. The result thus obtained is used to find the topological type of the real part of the Fano variety for the last rough projective class of real four-dimensional cubics which remained not investigated.

Rigid isotopy classification of real quadric line complexes and associated Kummer surfaces

The paper deals with rigid isotopy classes of three-dimensional real quadric line complexes and associated Kummer surfaces. We prove that there exist twelve rigid isotopy classes of real quadric line complexes and seven rigid isotopy classes of associated Kummer surfaces. Characteristics determining these rigid isotopy classes are given.

Topological classification of real three-dimensional cubics

The paper is devoted to finding topological types of nonsingular real three-dimensional cubics. It is proved that the following topological types exist: the projective space, the disjoint union of the projective space and the sphere, the projective space with handles whose number can vary from one to five. Along with these types, there is another topological type which is possibly distinct from those listed above, and this type is yet not completely described. A real cubic of this type is obtained from the projective space by replacing some solid torus in the space by another solid torus such that, under this replacement, the meridians of the first solid torus become parallels of the other solid torus, and conversely.

The Albanese map of the Fano surface of a real M-cubic threefold

The restriction to the set of real points of the Albanese map of the Fano surface of a real M-cubic threefold is considered. Some topological properties of this map are proved.