Characterizing rational homogeneous spaces via $\mathbb{C}^*$-actions

TL;DR

This paper characterizes certain smooth varieties with specific rational curve families and $ ext{C}^*$-actions, showing they are precisely the irreducible Hermitian symmetric spaces, thus linking geometric properties to symmetric space classification.

Contribution

It provides a characterization of irreducible Hermitian symmetric spaces via $ ext{C}^*$-actions and rational curves on smooth varieties with Picard number one.

Findings

Varieties with the specified properties are Hermitian symmetric spaces.

The presence of an equalized $ ext{C}^*$-action with an isolated extremal fixed point characterizes these spaces.

The results connect geometric actions to the classification of symmetric spaces.

Abstract

We study smooth varieties of Picard number one admitting a special dominating family of rational curves and an equalized $\mathbb{C}^*$-action. In particular we show that $X$ is a smooth variety of Picard number one with nef tangent bundle admitting an equalized $\mathbb{C}^*$-action with an isolated extremal fixed point if and only if $X$ is an irreducible Hermitian symmetric space.