A graphical calculus for 2-block Spaltenstein varieties

TL;DR

This paper introduces a graphical calculus for understanding the geometry and cohomology of 2-block Spaltenstein varieties, extending known results about Springer fibres and connecting to coloured cobordisms.

Contribution

It generalizes known results about Springer fibres to Spaltenstein varieties, develops a graphical calculus for their structure, and links to coloured cobordisms.

Findings

Developed a graphical calculus encoding the structure of Spaltenstein varieties.

Computed the cohomology of these varieties.

Established a connection with coloured cobordisms.

Abstract

We generalise statements known about Springer fibres associated to nilpotents with 2 Jordan blocks to Spaltenstein varieties. We study the geometry of generalised irreducible components (i.e. Bialynicki-Birula cells) and their pairwise intersections. In particular we develop a graphical calculus which encodes their structure as iterated fibre bundles with CP^1 as base spaces and compute their cohomology. At the end we present a connection with coloured cobordisms generalising a construction of Khovanov and Stroppel.