The Gysin homomorphism for homogeneous spaces via residues - PhD Thesis

TL;DR

This thesis develops residue formulas for the Gysin homomorphism in equivariant cohomology of partial flag varieties, generalizing symplectic geometry theorems to compute integrals via residues.

Contribution

It introduces residue-based formulas for Gysin homomorphisms in equivariant cohomology of flag varieties, extending symplectic geometry theorems to this setting.

Findings

Derived explicit residue formulas for Gysin homomorphisms

Extended Jeffrey--Kirwan localization to equivariant cohomology

Established push-forward formulas for types A, B, C, D flag varieties

Abstract

The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quotients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form \[\int_X \alpha = Res_{\mathbf{z}=\infty} \mathcal{Z}(\mathbf{z}, \mathbf{t}) \cdot \alpha(\mathbf{t}),\] for a certain residue operation and a map $\mathcal{Z}(\mathbf{z}, \mathbf{t})$, associated to the root system. The mentioned description relies on two following generalizations of theorems in symplectic geometry. We adapt to the equivariant setting (for torus actions) the Jeffrey--Kirwan nonabelian localization and a theorem relating the cohomology of symplectic reductions by a compact Lie group and by its maximal torus, following the approach by Martin. Applying the generalized theorems to certain contractible spaces acted upon by a product of unitary groups we derive the push-forward formula for partial flag varieties of types $A$, $B$, $C$ and $D$.