Andreas Cap, A. Rod Gover, Matthias Hammerl
Projective BGG Equations, Algebraic Sets, and Compactifications of Einstein Geometries
Preprint series: ESI preprints

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

53A20 Projective differential geometry

53B20 Local Riemannian geometry

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 new structures that are
analogues of Poincar\'e-Einstein manifolds.


Keywords: Projective differential geometry, compactifications, Poincare-Einstein manifolds, Einstein manifolds, conformal geometry, parabolic geometries