Duality and integrability on contact Fano manifolds

TL;DR

This paper investigates the classification of contact Fano manifolds, exploring their geometric structures and proposing methods to identify their properties, contributing to the conjecture that all such manifolds are homogeneous.

Contribution

It demonstrates how key Lie algebra structures can be derived from contact manifold geometry, advancing understanding of their classification and properties.

Findings

Lie algebra structures can be inferred from contact manifold geometry

Minimal rational curves are essential for understanding contact manifolds

Collected facts about bundles of projective lines may be independently useful

Abstract

We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from geometry of an arbitrary contact manifold. Minimal rational curves on contact manifolds (or contact lines) and their chains are the essential ingredients for our constructions. Along the way we collect several facts about bundles of projective lines admitting a contractible section, which might be of independent interest.