Spinors of real type as polyforms and the generalized Killing equation

We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kähler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module $$\Sigma $$ Σ of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of $$\text {AdS}_4$$ AdS 4 space-time.

Approximation of maps into spheres by piecewise-regular maps of class $$C^k$$

The aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise-regular maps of class $${\mathcal {C}}^k,$$ C k , where k is an arbitrary nonnegative integer.

Stratified-algebraic vector bundles

We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle ξ on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification {\mathcal{S}} of X such that the restriction of ξ to each stratum S in {\mathcal{S}} is an algebraic vector bundle on S. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles.

COMPLEX CYCLES AS OBSTRUCTIONS ON REAL ALGEBRAIC VARIETIES

LetYbe a compact nonsingular real algebraic variety of positive dimension. Then one can find a compact connected nonsingular real algebraic varietyX, which admits a continuous map intoYthat is not homotopic to any regular map. It is hard to determine the minimum dimension of such a varietyX. In this paper, new upper bounds for dimXare obtained. The main role in the constructions is played by complex algebraic cycles onY.