Isotrivial VMRT-structures of complete intersection type

TL;DR

This paper investigates the relationship between isotrivial varieties of minimal rational tangents and quasi-homogeneity in projective manifolds, showing that isotriviality does not always imply local flatness, especially in complete intersection cases.

Contribution

It demonstrates that isotrivial VMRT-structures are not necessarily locally flat and characterizes when Z-isotriviality implies flatness for complete intersections.

Findings

Isotrivial VMRT-structures are not always locally flat.

Counterexamples exist even with Picard number 1.

Many complete intersection cases satisfy the flatness condition.

Abstract

The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number 1. In fact, an isotrivial family of varieties of minimal rational tangents needs not be locally flat in differential geometric sense. This leads to the question for which projective variety Z, the Z-isotriviality of varieties of minimal rational tangents implies local flatness. Our main result verifies this for many cases of Z among complete intersections.