Balloon Animal Maps with Applications to the Cohomology of Hypertoric Hitchin Systems

Abstract

In this article we introduce the notion of a balloon animal map between broken toric varieties and construct several long exact sequences in cohomology related to them. We give a new proof of the deletion-contraction relation on hypertoric Hitchin systems of Dansco-Mcbreen-Shende and present some refinements of it. The end result is a formula for the Poincar\'e polynomial of any hypertoric Hitchin system associated to a graph with first Betti number 2 along with a recipe to calculate the Poincar\'e polynomial of any hypertoric Hitchin system, given knowledge of a finite number of base cases.