Euler-symmetric projective varieties

TL;DR

This paper introduces Euler-symmetric projective varieties, classifies them via algebraic symbol systems, and explores their geometric properties and relations to equivariant compactifications of vector groups.

Contribution

It provides a classification framework for Euler-symmetric projective varieties using symbol systems and analyzes their algebraic and geometric properties.

Findings

Euler-symmetric projective varieties are classified by symbol systems.

They are quasi-homogeneous and determined by fundamental forms.

Connections to equivariant compactifications of vector groups are established.

Abstract

Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the systems of fundamental forms at general points of projective varieties. We study relations between the algebraic properties of symbol systems and the geometric properties of Euler-symmetric projective varieties. We describe also the relation between Euler-symmetric projective varieties of dimension n and equivariant compactifications of the vector group G_a^n.