Algebraic varieties with additive group action

An additive action on an algebraic variety X is defined as a regular effective action of a commutative unipotent algebraic group with an open orbit. In a sense, varieties equipped with such actions are opposite to toric varieties, which contain an algebraic torus as an open dense orbit. We classify hypersurfaces that admit additive actions with a finite number of orbits.

In the affine case, for instance for horospherical varieties, actions of the additive group serve as a tool to study the automorphism group of a variety and its orbits structure. Note that the group of regular automorphisms of an affine variety is usually infinite-dimensional and not algebraic. However, by the absence of nontrivial additive group actions, the group of automorphisms may become small enough to admit a particular description. In this context, Perepechko and Zaidenberg conjectured that the neutral component of the automorphism group is a torus. We confirm this conjecture for several families of varieties. This talk is based on joint works with Sergey Gaifullin, Anton Shafarevich, and Alexander Chernov.