Ext Formulas For Intersection Cohomology Sheaves

Hey guys! Let's dive into the fascinating world of intersection cohomology sheaves! This topic sits at the intersection (pun intended!) of algebraic geometry, homological algebra, sheaf theory, and perverse sheaves. We're going to explore some conjectured formulas for computing the $\text{Ext}^1$ of these sheaves, which are super important for understanding the topology and geometry of singular spaces.

Setting the Stage: Stratified Spaces and Intersection Cohomology

First things first, let's set the stage. We're dealing with nice stratified spaces, which are basically spaces that can be broken down into smooth pieces (called strata) that fit together in a controlled way. Think of it like a building made of different blocks – each block is smooth, but the whole building might have corners and edges. We're assuming our spaces, denoted by $X$, have all strata connected and even dimensional. This simplifies things a bit and allows us to focus on the core concepts.

Now, what about intersection cohomology? Well, classical cohomology falls short when dealing with singular spaces (spaces with, like, pointy bits or self-intersections). It doesn't quite capture the topology in a satisfactory way. Intersection cohomology is a clever fix! It's a modified version of cohomology that's specifically designed to handle singularities. It was introduced by Goresky and MacPherson in the 1980s, and it's become a fundamental tool in geometric topology and representation theory.

The key ingredient in intersection cohomology is the intersection cohomology sheaf, often denoted as $IC_X$. This is a perverse sheaf, which is a special kind of object in sheaf theory that behaves well with respect to stratifications. Think of it as a way of packaging topological information about the space $X$ while being sensitive to its singular structure. The intersection cohomology groups are then obtained by taking the hypercohomology of this sheaf.

Why are we so interested in $\text{Ext}^1$ of intersection cohomology sheaves? Well, $\text{Ext}$ groups are a fundamental tool in homological algebra. They measure the ways in which objects can be extended or connected to each other. In this context, $\text{Ext}^1(IC_X, IC_X)$ essentially tells us about the self-extensions of the intersection cohomology sheaf. These self-extensions are deeply related to the geometry and topology of the space $X$. For example, they can provide information about deformations of the space or the existence of certain geometric structures.

Understanding the Ext groups involving intersection cohomology sheaves allows us to gain deeper insights into the structure of singular spaces. It helps us connect the local geometry around singularities with the global topology of the space. This is why finding formulas to compute these $\text{Ext}$ groups is such a hot topic!

Conjectured Formulas for $\text{Ext}^1$: A Deep Dive

Alright, let's get to the heart of the matter: the conjectured formulas for computing $\text{Ext}^1$ of intersection cohomology sheaves. The formulas I'm particularly interested in involve relating $\text{Ext}^1(IC_X, IC_X)$ to other topological invariants of the space $X$, specifically those arising from the stratification.

One prominent conjecture, which I'll call Conjecture A, relates $\text{Ext}^1(IC_X, IC_X)$ to the dimensions of certain cohomology groups of the links of strata. Let's break that down:

• Links of strata: Imagine zooming in really close to a point on a stratum. The link of that stratum is what you see