Rational curves and prolongations of G-structures

TL;DR

This paper explores the relationship between rational curves and G-structures on complex manifolds, aiming to generalize a previous result linking holomorphic G-structures and Lie algebra prolongation, but encounters a flaw in the proof.

Contribution

It attempts to extend a known theorem about holomorphic G-structures and Lie algebra prolongation to more general algebraic subgroups and complex manifolds with rational curves.

Findings

Initial proof had a flaw, preventing full generalization.

Highlights challenges in extending G-structure results.

Provides insights into the structure of complex manifolds with rational curves.

Abstract

In a joint work with N. Mok in 1997, we proved that for an irreducible representation $G \subset {\bf GL}(V),$ if a holomorphic $G$-structure exists on a uniruled projective manifold, then the Lie algebra of $G$ has nonzero prolongation. We tried to generalize this to an arbitrary connected algebraic subgroup $G \subset {\bf GL}(V)$ and a complex manifold containing an immersed rational curve, but the proposed proof had a flaw.