Characterizations of smooth projective horospherical varieties of Picard number one

TL;DR

This paper characterizes smooth projective horospherical varieties of Picard number one by their minimal rational tangents, establishing a biholomorphic equivalence under certain geometric conditions, and employs Lie algebra cohomology methods.

Contribution

It provides a new characterization of these varieties using minimal rational tangents and applies advanced Lie algebra cohomology techniques to study their geometric structures.

Findings

Biholomorphic equivalence under minimal rational tangent conditions

Computation of Lie algebra cohomology space of degree two

Vanishing of holomorphic sections of specific vector bundles

Abstract

Let $X$ be a smooth projective horospherical variety of Picard number one. We show that a uniruled projective manifold of Picard number one is biholomorphic to $X$ if its variety of minimal rational tangents at a general point is projectively equivalent to that of $X$. To get a local flatness of the geometric structure arising from the variety of minimal rational tangents, we apply the methods of $W$-normal complete step prolongations. We compute the associated Lie algebra cohomology space of degree two and show the vanishing of holomorphic sections of the vector bundle having this cohomology space as a fiber.