Equivariant Perverse Sheaves And Orbit Stratification
Hey everyone! Today, we're diving deep into the fascinating world of equivariant perverse sheaves and how they connect with orbit stratifications. This is a pretty cool area of math that brings together algebraic geometry, algebraic topology, representation theory, and sheaf theory. So, buckle up, and let's get started!
Introduction to Equivariant Perverse Sheaves
Let's start with the basics. Imagine you have a complex algebraic variety, which is just a fancy way of saying a geometric object defined by polynomial equations, and let's call it X. Now, suppose there's a connected algebraic group, which we'll call G, acting on X. Think of G as a group of symmetries or transformations that can be applied to X. The main focus of our discussion revolves around the interplay between the G-equivariant perverse sheaves on X and the underlying stratification of X induced by the group action.
In simpler terms, G-equivariant perverse sheaves are special kinds of mathematical objects (sheaves) that behave nicely with respect to the action of the group G. They carry information about the geometry of X and how it interacts with the group action. The key here is understanding that the G action naturally breaks X down into orbits, which are sets of points that can be transformed into each other by elements of G. This decomposition into orbits gives us a stratification of X, a way of organizing it into layers of different dimensions and complexities.
The forgetful functor plays a crucial role in this context. It's like a mathematical operation that strips away the G-equivariant structure of a perverse sheaf, leaving you with just a regular perverse sheaf on X. Understanding what information is preserved and what is lost by this functor is one of the central questions in this field. Specifically, we want to know how much of the G-equivariant structure can be recovered simply by looking at the underlying perverse sheaf and the stratification of X into G-orbits. This involves considering the relationship between the category of G-equivariant perverse sheaves and the category of perverse sheaves without the equivariance condition. The challenge lies in effectively bridging this gap and leveraging the orbit stratification to reconstruct the equivariant structure.
Moreover, this area has deep connections with representation theory, where groups and their actions are studied in a more abstract setting. The categories of perverse sheaves often have representation-theoretic interpretations, and understanding these connections can provide powerful tools for studying both the geometry of X and the representations of G. The decomposition theorem, a cornerstone result in the theory of perverse sheaves, plays a pivotal role in understanding how perverse sheaves decompose along strata in a stratified space. This theorem provides insights into the structure of the pushforwards of intersection cohomology complexes, which are fundamental objects in the study of singular spaces. The equivariant version of the decomposition theorem further enriches this understanding by incorporating the group action into the analysis, offering a more refined perspective on the geometry and topology of the space.
The Forgetful Functor and Its Implications
The forgetful functor, in essence, is a map that takes a G-equivariant perverse sheaf and
