Generically transitive actions on multiple flag varieties

Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity

We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.

Unipotent commutative group actions on flag varieties and nilpotent multiplications

Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.

Hassett-Tschinkel correspondence: Modality and projective hypersurfaces

B. Hassett and Yu. Tschinkel (1999) introduced a remarkable correspondence between generically transitive actions of a commutative unipotent algebraic group G and finite-dimensional local algebras. In this paper we develop Hassett-Tschinkel correspondence and calculate modality of generically transitive G-actions on projective spaces, classify actions of modality one, and characterize generically transitive G-actions on projective hypersurfaces of given degree. In particular, actions on degenerate projective quadrics are studied.

论旗簇平坦线性退化中的光滑轨迹

On the Smooth Locus in Flat Linear Degenerations of Flag Varieties

本文利用矩图技术给出旗簇平坦线性退化中光滑轨迹的组合刻画，推广了Feigin退化的相关结果，并提出用传递竞赛图判定固定点光滑性的新组合判据。

Generically multiple transitive algebraic group actions

With every nontrivial connected algebraic group $G$ we associate a positive integer ${\rm gtd}(G)$ called the generic transitivity degree of $G$ and equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$ admits an open orbit. We show that ${\rm gtd}(G)\leqslant 2$ (respectively, ${\rm gtd}(G)=\nobreak 1$) for all solvable (respectively, nilpotent) $G$, and we calculate ${\rm gtd}(G)$ for all reductive $G$. We prove that if $G$ is nonabelian reductive, then the above maximal $n$ is attained for…

Some results on equivariant contact geometry for partial flag varieties

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby's classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of $G$-equivariant vector bundles on $G/P$. A byprod…