Combinatorial aspects of bow varieties

Abstract

First introduced by Cherkis in theoretical physics, bow varieties form a rich family of symplectic varieties generalizing Nakajima quiver varieties. An algebro-geometric definition was later given by Nakajima and Takayama via moduli spaces of quiver representations. The main goal of this thesis is to study the torus equivariant cohomology of bow varieties. Our study is motivated by classical Schubert calculus and lays the foundation for a Schubert calculus for bow varieties where the underlying quiver is of finite type A. The crucial main mathematical tool we use is the theory of stable envelopes of Maulik and Okounkov. We show that this theory applies to bow varieties and study it with the main focus on explicit calculations. As a main result of this thesis we generalize a fundamental ingredient of classical Schubert calculus to the world of bow varieties: The Chevalley--Monk formula. Our generalization of this formula characterizes the multiplication of tautological divisors with respect to the stable envelope basis.