Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

TL;DR

This paper develops a framework for analyzing curved projective manifolds using normal tractor frames, leading to polynomial characterizations of solutions to natural PDEs and applications to Einstein-like geometries.

Contribution

It introduces normal tractor frames on curved projective manifolds, enabling polynomial descriptions of solutions to BGG equations and new insights into Einstein geometry compactifications.

Findings

Normal solutions are polynomial in generalized homogeneous coordinates.

Zero loci of solutions correspond to algebraic sets and stratifications.

Applications include constructing Einstein-like geometric structures.

Abstract

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivialisations arising from the special frames, normal solutions of classes of natural linear PDE (so-called first BGG equations) are shown to be necessarily polynomial in the generalised homogeneous coordinates; the polynomial system is the pull back of a polynomial system that solves the corresponding problem on the model. Thus questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects an orbit decomposition of the model. Applications include the construction of structures that are analogues of Poincare-Einstein manifolds.