Doubly indexed flag variety and fixed point set of a partial flag variety

TL;DR

This paper introduces a new class of smooth, projective varieties called doubly indexed flag varieties, and analyzes their structure and relation to fixed point sets of partial flag varieties under nilpotent endomorphisms.

Contribution

It defines doubly indexed flag varieties, describes their structure as iterated Grassmannian bundles, and relates them to fixed point sets of partial flag varieties stabilized by nilpotent endomorphisms.

Findings

Doubly indexed flag varieties are smooth and projective.

They can be described as iterated Grassmannian varieties.

Fixed point sets of partial flag varieties under nilpotent endomorphisms decompose into vector bundles.

Abstract

We define a variety of doubly indexed flags, this is a smooth, projective variety, and we describe it as an iterated over Grassmannian varieties. On the other hand, we consider the variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are vector bundles over varieties of the aforedmentioned type.