Post 4: Perverse Sheaves Part 1

25 Dec 2023

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In this post, I will give a very “high in the sky” overview of the subject of perverse sheaves. I won’t give many specifics - those will be for future posts.

For the more experienced reader, I’m going to be talking about sheaves of $R$-modules for a fixed commutative ring $R$; this is necessary for some of the things I will say, like the existence of the derived category of sheaves. I will also assume all sheaves are constructible – this is a necessary requirement for the theory of perverse sheaves.

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Necessary intuition

I will first give some brief intuition for two terms that pop up throughout this post. I will give more intuition and full definitions in the next post or two.

Sheaves

A sheaf is an object attached to a topological space. For the purposes of this post, you don’t need to know what a sheaf is.

The key fact about sheaves that you do need to know, at least conceptually, is that you can “glue” them together. First, if $\mc F,\mc G$ are sheaves on a space, then there is a notion of what it means for them to agree on an open subset. Now, consider the following setup:

• $X=U_1\cup U_2$ for two open subsets $U_1,U_2$,

• $\mc F_1,\mc F_2$ are sheaves on $U_1,U_2$ respectively, and

• $\mc F_1, \mc F_2$ agree¹ on $U_1\cap U_2$.

Then there is a unique sheaf $\mc F$ on $X$ that agrees with $\mc F_1$ on $U_1$ and agrees with $\mc F_2$ on $U_2$.

Gluing sheaves can be done with a bit more generality² than what I’ve presented, but hopefully this gets the idea across.

Categories galore

To even talk about what a perverse sheaf is, you need the language of derived categories. For the uninitiated, I will take a brief tour through the language of categories, abelian categories, and then derived categories.

Categories

Categories are collections of objects and morphisms between them. For instance, there is a category Grp whose objects are groups and whose morphisms are group homomorphisms. There is a category Top whose objects are topological spaces and whose morphisms are continuous functions. Sometimes mathematicians say “map” instead of “morphism”. One can also define isomorphisms in any category.

It is important to note that not all categories can be thought of as consisting of “sets with extra structure, and structure-preserving functions between them”, even though the familiar examples of categories are of this form.

There is a notion of two categories being equivalent. It is not quite the same as familiar notions of “being the same” in mathematics, like two groups being isomorphic or two topological spaces being homeomorphic. Instead, it means that the two categories are the same, possibly up to isomorphisms on the objects. For an illustrative³ example, there is a (unique) category with one object and one morphism, and it is equivalent to a category with two objects and four morphisms!

Abelian categories

Abelian categories are categories where one can “do homological algebra”. For instance, there is a zero object, every morphism has a kernel, quotients always exist, you can define exact sequences, you can define chain complexes and their cohomology, and so on. For the purpose of this and later posts, it is fine if you haven’t learned homological algebra before. The most familiar example of an abelian category is the category of $R$-modules for a fixed ring $R$. Notice that the categories of groups and topological spaces are not abelian – can you see why? It turns out that the category of sheaves is abelian.

An important feature of abelian categories⁴ are the simple objects. These are objects whose only subobjects⁵ are itself and 0.⁶ A similar notion you might be familiar with is that of simple groups. However, this isn’t the same notion! The two notions do coincide when you restrict to abelian groups, though.

Derived categories

Sometimes, one can take an abelian category $\mc A$ and construct its derived category, denoted $D(\mc A)$. The objects of $D(\mc A)$ are chain complexes in $\mc A$. You may have encountered chain complexes if you have studied homology or cohomology of topological spaces. If not, that’s fine. I just want to stress that objects of $D(\mc A)$ are not objects of $\mc A$. However, $\mc A$ does naturally sit inside $D(\mc A)$. One might call these the “individual $\mc A$-objects” in $D(\mc A)$; this phrasing shows up later.

One final thing to note is a certain abuse of linguistics concerning derived categories. Instead of saying “the derived category of the category of [blah] objects”, we say “the derived category of [blah] objects”.

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Perverse sheaves

So, what is a perverse sheaf? Well, it is not a sheaf, funny enough. This is an example of what some may call the “red herring principle”, so-called because red herrings are not herrings that are red. Instead, a perverse sheaf is a special kind of object in the derived category of sheaves.

Then, why use the name [adjective] sheaf? I don’t know for certain. However, one motivating reason is that perverse sheaves, like sheaves, can be glued together.

Another important fact is that perverse sheaves, like sheaves, form an abelian category.⁷ The simple perverse sheaves end up playing an important part of certain results connecting geometry and representation theory.

Let me briefly mention the notion of $G$-equivariant perverse sheaves for a group $G$. $G$-equivariant roughly means “compatible with a $G$-action”. In particular, if $X$ is the topological space you are studying, then it must have $G$ acting on it in order for you to define $G$-equivariant perverse sheaves on $X$.

A brief history lesson

There is an interesting story behind the mathematical development of perverse sheaves. I won’t get into it fully, but the idea is that an observation about solutions to differential equations ended up expanding into the so-called Riemann-Hilbert correspondence. One form of this correspondence says there is an equivalence of categories between the derived category of sheaves and the derived category of “regular holonomic algebraic $D$-modules”. I know, what a mouthful. The perverse sheaves are then the objects of the first category that correspond to the individual $D$-modules in the second category. I think it is truly remarkable that something as concrete as differential equations has evolved into such an abstract subject that bears no resemblance to its origin.

See also here (opens in new tab) for an explanation of how the adjective “perverse” came into use.

Applications

War, war, what are perverse sheaves good for? The reason that I study them is that they are fundamental to geometric representation theory. Essentially, that means the approach of studying representation theory with tools of algebraic geometry. Let me give some examples of major results in this area. Don’t worry if you don’t know some of the relevant key words; I also don’t know some of them.

1. The Springer correspondence⁸: Given a connected reductive algebraic group⁹ $G$, with Weyl group $W$ and nilpotent cone $\mc N$, there is an injection from the set of irreducible representations of $W$ to the set of simple $G$-equivariant perverse sheaves on $\mc N$.

2. The geometric Satake equivalence: For a given connected reductive algebraic group $G$, there is an equivalence of categories between $G(\C[[t]])$-equivariant perverse sheaves on the affine Grassmannian¹⁰ of $G$ and representations of the Langlands dual group of $G$.

3. Lusztig’s canonical bases¹¹: Given a quantum group, there is a collection of topological spaces called representation spaces, and the simple perverse sheaves on these representation spaces give a basis for the positive part of a quantum group.

So, hopefully you now have an appreciation for the utility and pervasiveness of perverse sheaves. I also hope that you will continue to learn about them in my next few posts. See you there!

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Footnotes: