Post 15: Perverse Sheaves Finale

01 Feb 2024

In the final post of my perverse sheaves series, I will discuss some important examples and properties of perverse sheaves.

I won’t split this into two parts, as I essentially just want to list some results. However, I will use some terms that I didn’t define in the intuition section of my posts, such as varieties. If you see the words “smooth” or “variety” here and don’t know what they mean, you can either ignore them or go back to the last post and learn them.

Also, if you know what varieties are and you didn’t read the last post, just note that I’m using the convention that variety means locally closed in the Zariski topology of $\C\P^n$ for some $n$.

Even though I only defined perverse sheaves for field coefficients, properties will hold for non-field coefficients unless otherwise specified. The coefficient ring $R$ still needs to have some “mild” assumptions on it, but I won’t belabor this point.

Let $X$ be a smooth irreducible variety of dimension $d$. Let $\mc L$ be a local system of finite type on $X$, and consider it as a complex in¹ degree $-d$. Then $\mc L$ is a perverse sheaf.

The perverse sheaves on a one-point space are exactly (up to quasi-isomorphism) the complexes $\mc F$ with $\mc F^i\neq 0$ iff $i=0$ and $\mc F^0$ a constructible sheaf. Recalling that sheaves on a point correspond to $R$-modules, it is easy to show that the constructible sheaves on a point correspond to finitely generated $R$-modules. So, succintly, the category of perverse sheaves on a point is equivalent to the category of finitely generated $R$-modules.
If $j:U\to X$ is the inclusion of an open subset, and if $\mc F\in D^b_c(X)$, then $\mc F$ is a perverse sheaf if and only if $j^*\mc F$ is a perverse sheaf. For clarification on this, see this footnote.²

If $i:Z\to X$ is the inclusion of a closed set, then $i_*$ sends perverse sheaves on $Z$ to perverse sheaves on $X$.
Let $\mc F$ be a perverse sheaf on $X$, and let $d=\dim X$. Then $H^i(\mc F)=0$ if $i>0$ or if $i< -d$.
If our coefficient ring $R$ is a field, then the Verdier duality functor $\bb D$ sends perverse sheaves to perverse sheaves.
If $f:X\to Y$ is a finite morphism³ and $\mc F\in D^b_c(X)$, then $\mc F$ is perverse if and only if $f_* \mc F$ is perverse.

Let the coefficient ring $R$ be a field, and let $\mc L$ be a local system of finite type, considered as a complex in degree 0. Also let $\mc F$ be a perverse sheaf. Then $R\H om(\mc L,\mc F)$ is a perverse sheaf.
Finally, here is a hard result that shows up in applications to representation theory. Let the coefficient ring $R$ be the field $\Q$ of rational numbers. Let $f:X\to Y$ be a proper morphism⁴. Then for all semisimple complexes⁵ $\mc F$, we have $f_*\mc F$ is a semisimple complex.

Hopefully these results give you an appreciation for perverse sheaves, after all the work we have gone through to define them.

What next?

If you want to continue learning about perverse sheaves, then of course you can delve into Pramod Achar’s book, which is probably the best source to learn about them. The biggest concept that you need for applications to represenation theory, and that I didn’t talk about, is equivariant perverse sheaves. Equivariant essentially means “respecting a group action”. For instance, if $f:X\to Y$ is a continuous map, and $X$ and $Y$ both have actions by a group $G$, then $f$ is equivariant if $f(gx)=gf(x)$ for all $g\in G$ and $x\in X$. Defining equivariant perverse sheaves turns out to be a little subtle, but nonetheless they show up a lot in applications to representation theory.

I don’t know if I will exposit on equivariant perverse sheaves, but I probably will talk about applications of perverse sheaves as I learn them in more detail. For now though, I will post about other topics.

Footnotes:

Form a chain complex $\mc F$ which has $\mc F^i \neq 0$ if and only if $i=-d$, and $\mc F^{-d}=\mc L$. ↩
Recall that $D^b_c(X)$ refers to the complexes in $D^b(X)$ whose cohomology sheaves are constructible. Furthermore, if $j:U\to X$ is the inclusion of an open subset, and $\mc F$ is a complex of sheaves on $X$, then $(j^*\mc F)^i = \mc F^i\mid_U$, where $F\mid_U(V) = F(V)$ for any open subset $V$ of $U$. ↩
A map $f:X\to Y$ of varieties is finite if for all closed subsets $Z$ of $X$ and all points $y\in Y$, we have $f(Z)$ is closed and $f^{-1}(y)$ is compact and finite. ↩
A map $f:X\to Y$ of varieties is proper if for all closed subsets $Z$ of $X$ and all points $y\in Y$, we have $f(Z)$ is closed and $f^{-1}(y)$ is compact. ↩
A semisimple complex $\mc F$ on $X$ is an element of $D^b_c(X)$ that is isomorphic to a finite direct sum of shifts of simple perverse sheaves. A shift of a complex $A$ is a complex $A[i]$ for $i\in\Z$, where $(A[i])^j = A^{i+j}$ and $d_{A[i]}^j = (-1)^jd_A^{i+j}$. A simple perverse sheaf is a perverse sheaf that is simple in the category of perverse sheaves; explicitly, a perverse sheaf $\mc F$ is simple if any non-zero monomorphism $\mc G\to \mc F$, with $\mc G$ a perverse sheaf, is an isomorphism. ↩