Twisted submanifolds of $${\mathbb {R}}^n$$

We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of $${\mathbb {R}}^n$$ R n determined by a set of smooth equations $$f^a(x)=0$$ f a ( x ) = 0 . We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) $$\star $$ ⋆ -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra $$\Xi _t$$ Ξ t of vector fields that are tangent to all the submanifolds that are level sets of the $$f^a$$ f a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted $$\Xi _t$$ Ξ t . We can consistently project a connection from the twisted $${\mathbb {R}}^n$$ R n to the twisted M if the twist is based on a suitable Lie subalgebra $${\mathfrak {e}}\subset \Xi _t$$ e ⊂ Ξ t . If we endow $${\mathbb {R}}^n$$ R n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra $${\mathfrak {k}}\subset {\mathfrak {e}}$$ k ⊂ e of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and $$\star $$ ⋆ -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $${\mathbb {R}}^3$$ R 3 and twisted hyperboloids embedded in twisted Minkowski $${\mathbb {R}}^3$$ R 3 [these are twisted (anti-)de Sitter spaces $$dS_2,AdS_2$$ d S 2 , A d S 2 ].

Complete scalar-flat Kähler metrics on affine algebraic manifolds

Let $$(X,L_{X})$$ ( X , L X ) be an n-dimensional polarized manifold. Let D be a smooth hypersurface defined by a holomorphic section of $$L_{X}$$ L X . We prove that if D has a constant positive scalar curvature Kähler metric, $$X {\setminus } D$$ X \ D admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $$n \ge 6$$ n ≥ 6 and there is no nonzero holomorphic vector field on X vanishing on D, (ii) the average of a scalar curvature on D denoted by $${\hat{S}}_{D}$$ S ^ D satisfies the inequality $$0< 3 {\hat{S}}_{D} < n(n-1)$$ 0 < 3 S ^ D < n ( n - 1 ) , (iii) there are positive integers $$l(>n),m$$ l ( > n ) , m such that the line bundle $$K_{X}^{-l} \otimes L_{X}^{m}$$ K X - l ⊗ L X m is very ample and the ratio m/l is sufficiently small.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

On a New Distance Approximation for an Implicit Polynomial Manifold Fitting

The application of a new approximate point-to-algebraic manifold distance formula is suggested to the geometric approach to curve fitting and surface reconstruction using implicit polynomial manifolds. A brief overview of the fitting methods features for implicit algebraic manifolds is given. To illustrate the possibilities of a new approximate point-to-manifold distance formula, the equidistant curves of the exact distance, Samson’s distance and the present formula are given. A four-step algorithm for implicit algebraic manifold fitting is proposed, using one of the algebraic fitting methods at the initial step, the present approximate formula for the distance finding to calculate the geometric criterion of approximation quality and an optimization method for updating the value of the vector of coefficients of the manifold. The first results of the proposed algorithm on test data are briefly characterized. In conclusion, the tasks and directions for further research are described.

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.

Tannakian Duality for Affine Homogeneous Spaces

Associated with any closed quantum subgroup and any index set I ⊂ {1,…,N} is a certain homogeneous space , called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds that can appear in this way.