Define the index of rigidity of $\mathcal F$ on $U^{\mathrm{an}}$ to be $\mathrm{rig}(\mathcal F,U^{\mathrm{an}})=\chi(X^{\mathrm{an}},(j^{\mathrm{an}})_*\mathrm{End}(\mathcal F))$.

Let $X$ be a separated $\mathbb C$-scheme locally of finite type. A partition of $X$ is a family of locally closed immersions $i_\alpha:Y_\alpha\hookrightarrow X_{\mathrm{red}}$, indexed by a set $A$, with each $Y_\alpha$ nonempty, smooth, and connected, such that:

1. $Y_\alpha\times_{X_{\mathrm{red}}}Y_\beta=\varnothing$ whenever $\alpha\ne\beta$;
2. the induced morphism $\coprod_\alpha Y_\alpha\longrightarrow X_{\mathrm{red}}$ is surjective.

The schemes $Y_\alpha$ are called the pieces of the partition. The partition is finite if $A$ is finite.

Let $X$ be a separated $\mathbb C$-scheme of finite type. A sheaf $\mathcal F$ of $\mathbb C$-vector spaces on $X^{\mathrm{an}}$ is algebraically constructible if there exists a finite partition $\{i_\alpha:Y_\alpha\hookrightarrow X_{\mathrm{red}}\}$ such that $(a_\alpha^{\mathrm{an}})^*\mathcal F$ is a finite-rank local system on each $Y_\alpha^{\mathrm{an}}$, where $a_\alpha:Y_\alpha\hookrightarrow X$ is the composite of $i_\alpha$ with $X_{\mathrm{red}}\hookrightarrow X$. This is an equivalent formulation of Katz's definition ⟦cite:Kat96⟧ .

On a smooth connected algebraic curve $U/\mathbb C$, a sheaf of $\mathbb C$-vector spaces $\mathcal F$ on $U^{\mathrm{an}}$ is called a middle extension sheaf if it is algebraically constructible and, for some (equivalently, every) nonempty Zariski open subset $i:V\hookrightarrow U$ such that $(i^{\mathrm{an}})^*\mathcal F$ is a local system on $V^{\mathrm{an}}$, the natural restriction morphism

$$\mathcal F\longrightarrow(i^{\mathrm{an}})_*(i^{\mathrm{an}})^*\mathcal F$$

is an isomorphism. A middle extension sheaf $\mathcal F$ is called irreducible if, for some (equivalently, every) nonempty Zariski open subset $i:V\hookrightarrow U$ on which $(i^{\mathrm{an}})^*\mathcal F$ is a local system, that local system is irreducible.

Let $j:U\hookrightarrow\mathbb P^1_{\mathbb C}$ be a nonempty Zariski open subset, and let $\mathcal F$ be an irreducible middle extension sheaf on $U^{\mathrm{an}}$. Choose a nonempty Zariski open subset $i:V\hookrightarrow U$ such that $(i^{\mathrm{an}})^*\mathcal F$ is an irreducible local system on $V^{\mathrm{an}}$. Define the index of rigidity of $\mathcal F$ as

$$\begin{aligned} \mathrm{rig}_{U^{\mathrm{an}}}(\mathcal F) &:=\mathrm{rig}((i^{\mathrm{an}})^*\mathcal F,V^{\mathrm{an}})\\ &=\chi\!\left((\mathbb P^1_{\mathbb C})^{\mathrm{an}},(j^{\mathrm{an}})_*(i^{\mathrm{an}})_*\mathrm{End}((i^{\mathrm{an}})^*\mathcal F)\right). \end{aligned}$$

For a morphism $f:X\to Y$ of separated $\mathbb C$-schemes of finite type, the direct image with proper support is the functor defined, for a sheaf $\mathcal F$ on $X^{\mathrm{an}}$ and an open subset $V\subseteq Y^{\mathrm{an}}$, by

$$\bigl((f^{\mathrm{an}})_!\mathcal F\bigr)(V)=\bigl\{s\in\Gamma((f^{\mathrm{an}})^{-1}(V),\mathcal F):\textrm{The map }\operatorname{Supp}(s)\hookrightarrow(f^{\mathrm{an}})^{-1}(V)\xrightarrow{f^{\mathrm{an}}}V\text{ is proper}\bigr\}.$$

The extraordinary inverse image $(f^{\mathrm{an}})^!$ is its right adjoint, characterized by isomorphisms

$$\operatorname{Hom}_{\mathrm D_c^b(X^{\mathrm{an}},\mathbb C)}\!\left(K,(f^{\mathrm{an}})^!L\right)\cong\operatorname{Hom}_{\mathrm D_c^b(Y^{\mathrm{an}},\mathbb C)}\!\left(\mathrm R(f^{\mathrm{an}})_!K,L\right),$$

natural in $K\in\mathrm D_c^b(X^{\mathrm{an}},\mathbb C)$ and $L\in\mathrm D_c^b(Y^{\mathrm{an}},\mathbb C)$. We denote $a:X\to\operatorname{Spec}\mathbb C$ as the structural morphism. The dualizing complex is

$$\omega_{X^{\mathrm{an}}}^{\bullet}:=(a^{\mathrm{an}})^!\mathbb C,$$

where $\mathbb C$ is concentrated at degree zero. For $K\in \mathrm D_c^b(X^{\mathrm{an}},\mathbb C)$, its Verdier dual ⟦cite:Mat11⟧ is

$$\mathbb D_{X/\mathbb C}K:=\mathrm R\mathcal{H}om_{\mathbb C}\!\left(K,\omega_{X^{\mathrm{an}}}^{\bullet}\right).$$

A complex $K\in \mathrm D_c^b(X^{\mathrm{an}},\mathbb C)$ is called semiperverse if

$$\mathrm{dim}\,\mathrm{Supp}(\mathcal H^i(K))\le -i$$

for all $i\in\mathbb Z$, and perverse if both $K$ and $\mathbb D_{X/\mathbb C}K$ are semiperverse.

Let $\mathcal D$ be a triangulated category with shift functor $[1]$. A $t$-structure on $\mathcal D$ is a pair of full subcategories $(\mathcal D^{\le0},\mathcal D^{\ge0})$, closed under isomorphisms, satisfying the following axioms. Put $\mathcal D^{\le n}:=\mathcal D^{\le0}[-n]$ and $\mathcal D^{\ge n}:=\mathcal D^{\ge0}[-n]$.

1. $\mathcal D^{\le0}\subseteq\mathcal D^{\le1}$ and $\mathcal D^{\ge1}\subseteq\mathcal D^{\ge0}$.

2. $\operatorname{Hom}_{\mathcal D}(A,B)=0$ whenever $A\in\mathcal D^{\le0}$ and $B\in\mathcal D^{\ge1}$.

3. Every $K\in\mathcal D$ admits a distinguished triangle $A\to K\to B\to A[1]$ with $A\in\mathcal D^{\le0}$ and $B\in\mathcal D^{\ge1}$.

Its heart is the full subcategory $\mathcal D^\heartsuit:=\mathcal D^{\le0}\cap\mathcal D^{\ge0}$.

On $\mathcal D=\mathrm D_c^b(X^{\mathrm{an}},\mathbb C)$, put

$${}^p\mathcal D^{\le0}:=\{K\in\mathcal D:K\text{ is semiperverse}\},$$

$${}^p\mathcal D^{\ge0}:=\{K\in\mathcal D:\mathbb D_{X/\mathbb C}K\text{ is semiperverse}\}.$$

These subcategories form the middle perverse $t$-structure ⟦cite:BBD82⟧ . The category of perverse sheaves is its heart:

$$\mathrm{Perv}(X^{\mathrm{an}},\mathbb C):={}^p\mathcal D^{\le0}\cap{}^p\mathcal D^{\ge0}.$$

Let $G$ be a smooth separated group scheme of finite type over $\mathbb C$, with multiplication $m:G\times_{\mathbb C}G\to G$ and projections $p_1,p_2:G\times_{\mathbb C}G\to G$. For $K,L\in\mathrm D_c^b(G^{\mathrm{an}},\mathbb C)$, their external tensor product is

$$K\boxtimes L:=(p_1^{\mathrm{an}})^*K\otimes_{\mathbb C}^{\mathbf L}(p_2^{\mathrm{an}})^*L\in\mathrm D_c^b((G\times_{\mathbb C}G)^{\mathrm{an}},\mathbb C).$$

Their compact convolution (or $!$-convolution) and $*$-convolution are respectively ⟦cite:Kat96⟧

$$K*_!L:=\mathrm R(m^{\mathrm{an}})_!(K\boxtimes L),$$

$$K*_*L:=\mathrm R(m^{\mathrm{an}})_*(K\boxtimes L).$$

The natural forget-supports morphism $\mathrm R(m^{\mathrm{an}})_!\to\mathrm R(m^{\mathrm{an}})_*$ induces a morphism $K*_!L\to K*_*L$.