Holomorphic geometric structures on Kähler–Einstein manifolds

Abstract

Type of publicationArticle ( HOLOMORPHIC GEOMETRIC STRUCTURES ON KÄHLER-EINSTEIN MANIFOLDS BENJAMIN MCKAYA b s t r ac t . We prove that the compact Kähler manifolds with c 1 ě 0 that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kähler manifolds with c 1 ă 0 that admit holomorphic cominiscule geometries are the locally Hermitian symmetric varieties.

The Hartogs extension problem for holomorphic parabolic and reductive geometries

“…Every 3-dimensional projective Klein geometry M = Γ\U for which U ⊂ P 3 contains a projective line is known [18]. On any connected complex manifold containing a rational curve, any holomorphic projective connection is flat [4] p. 9 corollary 4, has a Zariski dense family of deformations of the rational curve, and admits no holomorphic deformations, as the Schwarzian derivative has to vanish along the rational curves [24], so each of these Klein geometries is the unique holomorphic projective connection on each of those complex 3-folds.…”

“…Each Hermitian locally symmetric manifold has a flat holomorphic (X, G)-geometry given by the inclusion X * ⊂ X. This (X, G)-geometry is the unique holomorphic cominiscule geometry on that complex manifold [24]. Every compact complex manifold M with c 1 (M, T ) < 0 which admits a holomorphic cominiscule geometry is a locally Hermitian symmetric space [24].…”

“…This (X, G)-geometry is the unique holomorphic cominiscule geometry on that complex manifold [24]. Every compact complex manifold M with c 1 (M, T ) < 0 which admits a holomorphic cominiscule geometry is a locally Hermitian symmetric space [24]. There is one known exotic example [17] of a cominiscule geometry, exotic because it is defined on a smooth complex projective variety, but is not (i) X or (ii) a complex torus or (iii) a locally Hermitian symmetric manifold; in this one example, (X, G) = P 3 , P SL(4, C) .…”

Locally Homogeneous Holomorphic Geometric Structures on Projective Varieties

“…Its developing map to P 2 intersects Q unless M is a ball quotient. Each ball quotient has a unique holomorphic projective connection, which is flat [46], and hence it is an (X, G)-structure for this (X, G) since it develops to the ball in the projective plane.…”