A characterization of irreducible Hermitian symmetric spaces of tube type by $\mathbb{C}^{*}$-actions

TL;DR

This paper characterizes irreducible Hermitian symmetric spaces of tube type among smooth projective varieties with Picard number one using special $ ext{C}^*$-actions that exhibit Euler type behavior at pairs of points.

Contribution

It establishes a new characterization of Hermitian symmetric spaces of tube type via $ ext{C}^*$-actions with Euler type properties at pairs of points.

Findings

Characterization of Hermitian symmetric spaces of tube type using $ ext{C}^*$-actions.

Equivalence between geometric structure and existence of specific $ ext{C}^*$-actions.

Provides a criterion to identify these spaces based on $ ext{C}^*$-symmetries.

Abstract

A $\mathbb{C}^{*}$-action on a projective variety $X$ is said to be of Euler type at a nonsingular fixed point $x$ if the isotropy action of $\mathbb{C}^{*}$ on $T_{x}X$ is by scalar multiplication. In this paper, it's proven that a smooth projective variety of Picard number one $X$ is isomorphic to an irreducible Hermitian symmetric space of tube type if and only if for a general pair of points $x,y$ on $X$, there exists a $\mathbb{C}^{*}$-action on $X$ which is of Euler type at $x$ and its inverse action is of Euler type at $y$.