Euler-symmetric projective toric varieties and additive actions

Abstract

Let Ga be the additive group of the field of complex numbers ℂ. We say that an irreducible algebraic variety X of dimension n admits an additive action if there is a regular action of the group Gan=Ga×…×Ga (n times) on X with an open orbit. In 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification of Euler-symmetric varieties and proved that any Euler-symmetric variety admits an additive action. In this paper we show that in the case of projective toric varieties the converse is also true. More precisely, a projective toric variety admitting an additive action is an Euler-symmetric variety with respect to any linearly normal embedding into a projective space. Also we discuss some properties of Euler-symmetric projective toric varieties.