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

A ${\mathbb{C}}^{\ast }$-action on a projective variety $X$ is said to be of Euler type at a nonsingular fixed point $x$ if the isotropy action of ${\mathbb{C}}^{\ast }$ 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 ${\mathbb{C}}^{\ast }$-action on $X$ which is of Euler type at $x$ and its inverse action is of Euler type at $y$.