\section{Perverse Sheaves (Christian Klevdal)}

The original reference for this topic the paper by Beilinson-Bernstein-Deligne (BBD) "Faisceux pervers". The name perverse sheaf is somewhat misleading since they are not honest sheaves, rather they live in $D^b_c(X)$, the bounded derived category of constructible sheaves on a topological space. In what follows, we will either take $X$ to be a variety over the complex numbers in the classical (analytic) topology, or a variety over any perfect field $k$, in the \'etale topology.  

Perverse sheaves are the heart of a non-standard t-structure on $D^b_c(X)$. Recall that for an abelian category $\mc{A}$, the standard t-structure on $D(\mc{A})$ takes $D^{\leq0}$, respectively $D^{\geq 0}$ to be the complexes $K$ with $H^i(K) = 0$ for $i > 0$, respectively $i < 0$. Before getting down to the specific case of perverse sheaves, we recall the definition and some generalities about t-structures.

%Definition of t-structure
\begin{defn}
A t-structure on a triangulated category $\mc{D}$ is a pair $( \mc{D}^{\leq0}, \mc{D}^{\geq0})$ of full subcategories where (with the notation $\mc{D}^{\geq n} = \mc{D}^{\geq 0}[-n]$ and likewise for $\mc{D}^{\leq n}$) we have:

\begin{enumerate}
\item $\mc{D}^{\leq 0} \subset \mc{D}^{\leq 1}$ and $\mc{D}^{\geq 1} \subset \mc{D}^{\geq 0}$. 
\item $\mathrm{Hom}_{\mc{D}}(\mc{D}^{\leq 0}, \mc{D}^{\geq 1}) = 0$. 
\item For all $X \in \mc{D}$, there exists a distinguished triangle 
	\[ Y \to X \to Z \to Y[1] \]
with $Y \in \mc{D}^{\leq 0}$ and $Z \in \mc{D}^{\geq 1}$. 
\end{enumerate}
\end{defn}

Recall that in the case of $D(\mc{A})$ and the standard t-structure we had truncation functors 
\begin{align*}
	\tau^{\leq n} \colon D(\mc{A}) &\to D(\mc{A})^{\leq n} \quad && K \mapsto \quad  \cdots \to K^{n-1} \to \ker(d^n) \to 0 \to \cdots \\
	\tau^{\geq n} \colon D(\mc{A}) &\to D(\mc{A})^{\geq n} \quad && K \mapsto \quad \cdots \to 0 \to \mathrm{coker}(d^n) \to K^{n+1} \to \cdots 
\end{align*}
and we used these functors to show the existence of the distinguished triangle in axiom 3 of a t-structure. Our next step is to show that given any t-structure, such truncation functors always exist. 

%Existence of truncation functors
\begin{prop}
Let $(\mc{D}^{\leq0}, \mc{D}^{\geq0})$ be a t-structure on a triangulated category $\mc{D}$. Then
\begin{enumerate}
\item There exists a left adjoint $\tau^{\geq n}$ for the inclusion $\mc{D}^{\geq n} \to \mc{D}$ and a right adjoint $\tau^{\leq n}$ for the inclusion $\mc{D}^{\leq n} \to \mc{D}$. 
\item For all $X \in \mc{D}$ there exists a unique $\delta_X \colon \tau^{\geq n+1}X \to \tau^{\leq n}X[1]$ such that 
	\[ \tau^{\leq n}X \to X\to \tau^{\geq n}X \xrightarrow{\delta_X} \tau^{\leq n}X[1] \]
is a distinguished triangle (the maps to and from $X$ are the canonical maps from adjunction).
\end{enumerate}  
\end{prop}
\begin{proof}
First, we may assume $n = 0$. By (3) from the definition of a t-structure, there exists some distinguished triangle 
	\[ X_{\leq 0} \to X \to X_{\geq 1} \to X_{\leq 0}[1] \]
with $X_{\leq 0} \in \mc{D}^{\leq 0}$ and $X_{\geq 1} \in \mc{D}^{\geq 1}$. If $Y$ is in $\mc{D}^{\leq 0}$ we apply $\mathrm{Hom}(Y, \cdot)$ to the above exact triangle and get an exact sequence 
	\[\mathrm{Hom}(Y,X_{\geq 1}[-1]) \to \mathrm{Hom}(Y, X) \to \mathrm{Hom}(Y,X_{\leq 0}) \to \mathrm{Hom}(Y, X_{\geq 1}) \]
But we know that the first and last terms of the sequence are zero from properties (1), (2) of a t-structure. Defining $\tau^{\leq 0}X = X_{\leq 0}$ we get the required isomorphism. However, we need to check that this definition of $\tau^{\leq 0}$ gives a well defined functor, and that $\tau^{\leq 0}$ is the right adjoint of inclusion (i.e. that the isomorphisms given above are functorial). It is sufficient to show for each map $X \to X'$ in $\mc{D}$ that there is a unique factorization 
\begin{center}
\begin{tikzcd}
\tau^{\leq 0}X \arrow{d} \arrow[dotted]{r}{\exists !} & \tau^{\leq 0}X' \arrow{d} \\
X \arrow{r} & X' 
\end{tikzcd}
\end{center}
To see the unique factorization, consider the exact triangle $\tau^{\leq 0}X' \to X' \to X'_{\geq 1} \to (\tau^{\leq 0}X')[1]$. Applying $\mathrm{Hom}(\tau^{\leq 0} X, \cdot)$ we get an exact sequence 
	\[ \mathrm{Hom}(\tau^{\leq 0} X,X'_{\geq 1}[-1]) \to \mathrm{Hom}(\tau^{\leq 0} X, X') \to \mathrm{Hom}(\tau^{\leq 0} X,\tau^{\leq 0}X') \to \mathrm{Hom}(\tau^{\leq 0} X, X'_{\geq 1}) \]
Again the first and last terms vanish. Thus the middle arrow is an isomorphism and we get a unique factorization as required. If we define $\tau^{\geq 1}X = X_{\geq 1}$ a similar proof shows that this is a well defined functor that is left adjoint to inclusion. 

The second claim follows from the following lemma.
\end{proof}

%Lemma on connection homomorphisms
\begin{lem}
Suppose we are given maps $\delta_1, \delta_2$ such that $A \to X \to B \xrightarrow{\delta_i} A[1]$ is an exact triangle for $i=1,2$. Then $\delta_1 = \delta_2$ if $\mathrm{Hom}(A[1], B) = 0$. 
\end{lem}
\begin{proof}
Consider the diagram
\begin{center}
\begin{tikzcd}
A \arrow{r}{a} \arrow{d}{id} & X \arrow{r}{b} \arrow{d}{id} & B \arrow{r}{\delta_1} \arrow[dotted]{d}{f} & A[1] \arrow{d}{id} \\
A \arrow{r}{a} & X \arrow{r}{b} & B \arrow{r}{\delta_2} & A[1] 
\end{tikzcd}
\end{center}
The existence of the dotted arrow making this a morphism of triangles follows from the axioms of a triangulated category. Our aim is to show that $f = id_B$. Now we have that $fb = b$ or equivalently $(id_B - f)b = 0$. Applying $\mathrm{Hom}(\cdot, B)$ to the top row we get an exact sequence 
	\[ 0 = \mathrm{Hom}(A[1],B) \to \mathrm{Hom}(B,B) \to \mathrm{Hom}(X,B) \]
Clearly $f- id_B$ maps to zero in the last Hom set. Since the map is injective, we conclude that $f = id_B$ as required. 
\end{proof}

\begin{defn}
Let $\mc{D}$ be a triangluated category and $(\mc{D}^{\leq 0}, \mc{D}^{\geq 0})$ a t-structure on $\mc{D}$. The heart of this t-structure is $\mc{D}^\heartsuit = \mc{D}^{\leq 0}\cap\mc{D}^{\geq 0}$. 
\end{defn}

We can define cohomology a cohomology functor $H^0 \colon \mc{D} \to \mc{D}^\heartsuit$ with respect to this t-structure to be the composition $\tau^{\leq 0} \circ \tau^{\geq 0} (\simeq \tau^{\geq 0} \circ \tau^{\leq 0}$. Likewise, we define $H^n = H^0 \circ [n]$. 

\begin{exmpl}
If $\mc{A}$ is an abelian category we can give $D(A)$ the standard t-structure (where $\mc{D}^{\leq 0}$, resp. $\mc{D}^{\geq 0}$, consists of those complexes with zero cohomology in degree $i > 0$, resp. $i < 0$). In this case, the heart of this t-structure can be naturally identified with $\mc{A}$ (by consider objects as chain complexes concentrated in degree 0); and the cohomology functor is just the usual cohomology of a chain complex. 
\end{exmpl}

Note that in the above example, the heart of the t-structure is an abelian category. Surprisingly, this turns out to be true in general.

%The heart is an abelian category
\begin{thm}
(BBD) For any t-structure on a triangulated category $\mc{D}$, the heart $\mc{D}^\heartsuit$ is an abelian category. 
\end{thm}

The next natural question is what can we say about $\mc{D}^\heartsuit$? One thing that holds is that for $A, B \in \mc{D}^\heartsuit$ one has $\mathrm{Ext}^1_{\mc{D}^\heartsuit}(A, B) \cong \mathrm{Hom}(A, B[1])$. We caution that the analogue does not hold for higher exts. 

%Triangulated category not equivalent to derived category of the heart
\begin{exmpl}
Suppose $X$ is a (reasonable) topological space, and let
\[ 
\mc{D} = D_\text{loc}(X) = \{ K \in D(X) \colon H^{i}(K) \text{ is locally constant for all } i \}.
\] 
The standard t-structure on $D(X)$ induces a t-structure on $\mc{D}$. If $X$ is connected and simply connected, then locally constant sheaves are constant, hence $\mc{D}^\heartsuit$ is equivalent to the category Ab of abelian groups. Thus if $n > 1$
	\[ \mathrm{Ext}^n_{\mc{D}^\heartsuit}(\underline{\ints}, \underline{\ints}) \cong \mathrm{Ext}^n_{\text{Ab}}(\ints, \ints) = 0 \]
However, 
	\[ \mathrm{Hom}_{\mc{D}}(\underline{\ints}_X, \underline{\ints}_X[n]) = H^0 \mathrm{RHom}(\underline{\ints}_X, \underline{\ints}_X[n]) = H^0 \mathrm{R}\Gamma(X, \underline{\ints}_X[n]) = H^n(X, \ints) \]
So if $X$ is simply connected but has higher cohomology (eg, $X= S^2$), we see that $D(\mc{D}^\heartsuit)$ is not equivalent to $D_\text{loc}(X)$. 
\end{exmpl}

However, in the cases we are interested in, we have the following

\begin{thm}
(Nori) If $X/k$ is a quasi-projective variety, then $D^b_c(X)$ is equivalent to the bounded derived category of the abelian category of constructible sheaves on $X$. (Beilinson) These are equivalent to the bounded derived category of the abelian category of perverse sheaves on $X$. 
\end{thm}
Both of these assertions have geometric content!

Before defining the perverse t-structure, we make a couple more general definitions. First recall (excercise):
\begin{thm}
For any t-structure $(\mc{D}^{\leq 0}, \mc{D}^{\geq 0})$ on a triangulated category $\mc{D}$, the cohomology functor $H^0 \colon \mc{D} \to \mc{D}^\heartsuit$ is a cohomological functor (meaning it takes distinguished triangles to long exact sequences). 
\end{thm}

\begin{defn} Lef $F \colon \mc{D}_1 \to \mc{D}_2$ be exact functors of triangulated categories (meaning that $F$ commutes with the shift and preserves distinguished triangles). Suppose we have t-structures $(\mc{D}_i^{\leq 0}, \mc{D}_i^{\geq 0})$ on $\mc{D}_i$ for $i = 1,2$. Then we say that $F$ is t-left exact if $F(\mc{D}_1^{\geq 0}) \subset \mc{D}_2^{\geq 0}$ and that $F$ is t-right exact if $F(\mc{D}_1^{\leq 0}) \subset \mc{D}_2^{\leq 0}$. We say that $F$ is t-exact if it is both t-left and t-right exact. 
\end{defn}

In the setting of the definition (with out any assumptions on the t-exactness of $F$), we define a functor $F^\heartsuit \colon \mc{D}_1^\heartsuit \to \mc{D}_2^\heartsuit$ by the composition 
	\[ \mc{D}_1^\heartsuit \subset \mc{D}_1 \xrightarrow{F} \mc{D}_2 \xrightarrow{H^0} \mc{D}_2^\heartsuit \]
As a remark, in BBD, $F^\heartsuit$ is denoted ${}^pF$. We may later lapse into this notation. 

Before defining the perverse t-structure (whose heart will be the category of perverse sheaves) we begin with a little motivation. The source of the theory is intersection cohomology $IH^\ast(X)$ which for a variety $X$ is defined as $H^\ast(X, IC_X)$ where $IC_X$ is a particular perverse sheaf on $X$ called the intersection complex. Here are some of the good features of $IH^\ast(X)$. 
\begin{enumerate}
\item $IH^\ast(X) = H^\ast(X)$ if $X$ is smooth.
\item $IH^\ast(X)$ always satisfies Poincare duality (if $X$ is not projective, you need the compact supports version--we'll see this later).
\item For any projective $X$, $IH^m(X)$ is pure of weight $m$, either in the sense of Hodge theory ($k=\mathbb{C}$) or Frobenius eigenvalues ($k= \mathbb{F}_q$). 
\end{enumerate}

Let's elaborate on this last point. In the case that $X/\cmplx$ is smooth and projective, $H^m(X, \rats)$ is naturally endowed with a pure Hodge structure of weight $m$. However if $X$ is projective but not smooth, $H^m(X, \rats)$ has a mixed Hodge structure of weights $\leq m$. In the case that $X/\mathbb{F}_q$ is smooth and projective, then the action of Frobenius on $H^m_\text{\'et}(X_{\mathbb{F}_q}, \rats_\ell)$ is pure of weight $m$, meaning all eigenvalues are algebraic numbers with archimedean absolute values $q^{m/2}$. If we don't assume $X$ is smooth, then the \'etale cohomology is mixed with weights $\leq m$. What the final point above is saying is that when $X$ is projective, but possibly singular, $IH^m(X)$ is pure of weight $m$ in the appropriate sense. 

We now define the perverse t-structure. In this setting, let $k$ be a perfect field, $X/k$ a separated scheme of finite type over $k$ (which we may assume is reduced). We write $D(X)$ for $D^b_c(X)$ the subcategory of the bounded derived category of sheaves on $X$ with constructible cohomology.

\begin{defn}
We define full subcategories of $D(X)$ by 
	\[ {}^pD^{\leq 0}(X) = \{ K \in D(X) \colon \mathrm{dim} (\mathrm{supp} H^iK) \leq -i, \forall i \} \]
	\[ {}^pD^{\geq 0}(X) = \{ K \in D(X) \colon \mathrm{dim} (\mathrm{supp} H^i(\mathbb{D}_XK)) \leq -i, \forall i \} \]
(Recall that $\mathbb{D}_XK = \mathrm{R}\mc{H}om(K, \pi^!\rats_\ell)$ for $\pi \colon X \to \mathrm{Spec} k$).
\end{defn}
This is the ``self-dual'' (for reasons we will see) or ``middle'' perversity. There are variants where one changes the numerics of the above definition, but this is the version of central importance in algebraic geometry.
\begin{exmpl} 
If $X$ is smooth, dimension $d$ and $\mc{F}$ a local system on $X$ then $\mc{F}[d] \in {}^pD^{\geq 0} \cap {}^pD^{\leq 0}$. To see that it is in ${}^pD^{\leq 0}$ note that $H^i(\mc{F}[d])$ is non zero only in degree $d$, where it has full support. The dual statement will follow from Lemma \ref{smooth}
\end{exmpl}

\begin{defn} 
The category of perverse sheaves on $X$ is $\mathrm{Perv}(X) = {}^pD^{\geq 0} \cap {}^pD^{\leq 0}$. 
\end{defn}

\begin{thm}
$({}^pD^{\leq 0}(X), {}^pD^{\geq 0}(X))$ is a t-structure on $D(X)$ with heart $\mathrm{Perv}(X)$. 
\end{thm}

This theorem is proven by induction on the dimension of $X$, using the BBD technique of gluing abstract t-structures. In addition to this formalism, we will have to know what happens in ``the smooth case.'' We define $D_\text{loc}(X) = \{ K \in D(X) \colon H^i(X) \text{ is locally constant } \forall i \}$, and we next show that when $X$ is smooth, the perverse t-structure on $D_{\text{loc}}(X)$ is just a translate of the standard t-structure. 

\begin{prop}
If $U$ is smooth of dimension $d$ then 
\[ ({}^pD^{\leq 0}(U)\cap D_\text{loc}(U), {}^pD^{\geq 0}(U)\cap D_\text{loc}(U)) = (D^{\leq -d}_\text{loc}(U), D^{\geq -d}_\text{loc}(U)) \]
(a shift of the standard t-structure). 
\end{prop}
\begin{proof}
For $K \in D_\text{loc}(U)$ if $H^{-i}K \neq 0$ then $\mathrm{supp} H^{-i}K = U$ since the cohomology is a local system. Thus the condition $K \in {}^pD^{\leq 0}(U)\cap D_\text{loc}(U)$ is equivalent to $H^iK = 0$ for all $i > -d$. Hence ${}^pD^{\leq 0}(U)\cap D_\text{loc}(U) = D^{\leq -d}_\text{loc}(U)$. On the other hand $K \in {}^pD^{\geq 0}(U)$ if and only if $\mathbb{D}_UK \in {}^pD^{\leq 0}(U)$ which is equivalant to saying $H^i(\mathbb{D}_UK) = 0$ for all $i > -d$. The following lemma shows that this is equivalent to having $H^jK  = 0$ for all $j < -d$. 
\end{proof}

\begin{lem}\label{smooth}
If $U$ is smooth of dimension $d$ and $K \in D_\text{loc}(U)$ then for all $i$ 
	\[ H^i(\mathbb{D}_UK) \cong H^{-i-2d}(K)^\vee(d) \]
(where the (d) is a Tate twist in case we are working with \'etale cohomology).
\end{lem}
\begin{proof}
Proceed by induction on $|\{i \colon H^iK \neq 0 \}|$. Given $K$ let $n$ be the maximal index such that $H^nK \neq 0$. We get a distinguished triangle 
	\[ \tau^{\leq n-1}K \to K \to \tau^{\geq n}K \xrightarrow{+1} \]
Since $n$ is maximal, we know that $\tau^{\geq n}K \cong H^nK[-n]$. Write $\mc{F} = H^nK$. Now we apply $R\mc{H}om(\cdot, \mc{K}_U) = \mathbb{D}_U$ and take the long exact sequence of the resulting distinguished triangle. Since $U$ is smooth we have $\mc{K}_U = \rats_{\ell}[2d](d)$ and the long exact sequence is 
	\[ \cdots \to H^i(R\mc{H}om(\mc{F}[-n], \rats_\ell[2d])) \to H^i(R\mc{H}om(K, \rats_\ell[2d])) \to H^i(R\mc{H}om(\tau^{\leq n-1}K, \rats_\ell[2d])) \to \cdots \]
Note that the first term can be can be computed as
	\[ H^{i + n +2d} R\mc{H}om(\mc{F}, \rats_\ell) = \begin{cases} 0 &\text{ if } i \neq -n-2d \\ \mc{F}^\vee & \text{ if } i = -n - 2d \end{cases} \]
since $\mc{F}$ is locally constant (if $\mc{F}$ is constant, i.e. $\rats_\ell$, then $\mc{H}om(\rats_\ell, \cdot)$ is the identity functor so there are no higher derived functors; since $\mc{F}$ is locally constant we can reduce to this case). This implies that 
	\[ 
	H^iR\mc{H}om(K, \rats_\ell[2d]) = 
	\begin{cases} 	\mc{F}^\vee &\text{ if } i = -2d -n \\ 
			H^i(\mathbb{D}_U\tau^{\leq n-1}K) &\text{ if } i > -2d - n \\
			0 &\text{ if } i < -2d-n 
	\end{cases}
	\]
(Check the details as an exercise. For instance, 
\[
H^i R\mc{H}om(\tau^{\leq n-1}K, \rats_{\ell}[2d])=H^0 R\mc{H}om(\tau^{\leq n-1}K, \rats_{\ell}[2d+i]),
\]
so if $i \leq -2d-n$, i.e. $-2d-i \geq n$, then this group vanishes for degree reasons.)
The lemma now follows by applying the induction hypothesis to $\tau^{\leq n-1} K$. 
\end{proof}
The following important lemma gives a way to describe ${}^p D^{\leq 0}(X)$ and ${}^p D^{\geq 0}(X)$ in terms of restricting to a decomposition $X= U \sqcup Z$. It motivates the abstract glueing procedure for t-structures to be described in the next section.
\begin{lem}
 Let $X/k$ be separated of finite type. Let $j \colon U \to X$ be an open imbedding with closed complement $i \colon Z \to X$, and let $K \in D(X)$. then
\begin{itemize}
 \item $K \in {}^p D^{\leq 0}(X)$ if and only if $j^*K \in {}^p D^{\leq 0}(U)$ and $i^*K \in {}^p D^{\leq 0}(Z)$; and
 \item $K \in {}^p D^{\geq 0}(X)$ if and only if $j^*K \in {}^p D^{\geq 0}(U)$ and $i^!K \in {}^p D^{\geq 0}(Z)$.
\end{itemize}

\end{lem}
\begin{proof}
 Since $i^*$ and $j^*$ are exact for the standard t-structure, $\mathrm{Supp}(H^n K)= \mathrm{Supp}(j^*H^n(K)) \sqcup \mathrm{Supp}(i^*H^n(K))$. The first item follows. For the second, $K \in {}^p D^{\geq 0}(X) \iff \mathbb{D}_X(K) \in  {}^p D^{\leq 0}(X)$, which holds (by the first part) if and only if $j^*\mathbb{D}_X(K) \in  {}^p D^{\leq 0}(U)$ and $i^*\mathbb{D}_X(K) \in  {}^p D^{\leq 0}(Z)$. Since $j^* \mathbb{D}_X \simeq \mathbb{D}_U j^!$ and $i^*\mathbb{D}_X \simeq \mathbb{D}_Z i^!$, the second item follows from the definitions of the perverse ``t-structures'' (not yet) on $U$ and $Z$.
\end{proof}

