Minimal rational curves and 1-flat irreducible G-structures

TL;DR

This paper explores 1-flat irreducible G-structures in algebraic geometry, showing they are locally symmetric or flat under certain conditions, using Cartan connections on minimal rational curves.

Contribution

It extends the study of 1-flat G-structures to algebraic geometry via VMRTs, proving local symmetry in high dimensions and local flatness in specific cases.

Findings

Structures are locally symmetric when dimension ≥ 5.

VMRTs are locally flat without 1-flatness assumption.

Cartan connections are used on spaces of minimal rational curves.

Abstract

1-flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachh\"ofer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.