Sheaves, derived and perverse

Let $X,Y$ be topological spaces and $f:X\to Y$ a continuous map. We let $\Shv(X)$ be the category of sheaves on $X$, $D(\Shv(X))$ the derived category of sheaves on $X$, and $D_b(\Shv(X))$ the bounded variant. Recall that $D(\mathcal A)$ for an abelian category $\mathcal A$ is constructed first by taking $C(\mathcal A)$, the category of cochains of elements of $\mathcal A$, quotienting by chain homotopy, then quotienting by all acylic chains.

Remark: Let $\mathcal F\in \Shv(X)$. Recall:

• a section of $\mathcal F$ is an element of $\mathcal F(U)$ for some $U\subseteq X$,
• a germ of $\mathcal F$ at $x\in X$ is an equivalence class in $\{s\in \mathcal F(U)\ :\ U\owns x\}/\sim_x$,
• $s\sim_x t$ iff every neighborhood $W$ of $x$ in $U\cap V$ has $s|_W = t|_W$, for $s\in \mathcal F(U)$, $t\in \mathcal F(V)$,
• the support of the section $s\in\mathcal F(U)$ is $\supp(s) = \{x\in U\ :\ s \nsim_x 0\}$,
• the support of the sheaf $\mathcal F$ is $\supp(\mathcal F) = \{x\in X\ :\ \mathcal F_x\neq 0\}$.

Definition: The map $f$ induces functors between categories of sheaves, called
\[\begin{array}{r r c l}
\text{direct image} & f_*\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & (V\mapsto \mathcal F(f^{-1}(V))),\\[15pt]
\text{inverse image} & f^*\ :\ \Shv(Y) & \to & \Shv(X), \\
& (V\mapsto \mathcal G(V)) & \mapsto & \text{sh}\left(U\mapsto \text{colim}_{V\supseteq f(U)} \mathcal G(V)\right),\\[15pt]
\text{direct image with compact support} & f_!\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & \left(V\mapsto \left\{ s\in\mathcal F(f^{-1}(V))\ :\ f|_{\supp(s)} \text{ is proper}\right\}\right).
\end{array}\]

Above we used that $f:X\to Y$ is proper if $f^{-1}(K)\subseteq X$ is compact, for every $K\subseteq Y$ compact. Next, recall that a functor $\varphi:\mathcal A\to \mathcal B$ induces a functor $R\varphi:D(\mathcal A)\to D(\mathcal B)$, called the (first) derived functor of $\varphi$, given by $R\varphi(A^\bullet) = H^1(\varphi(A)^\bullet)$.

Remark: Each of the maps $f_*,f^*,f_!$ have their derived analogues $Rf_*, Rf^*,Rf_!$, respectively. For reasons unclear, $Rf_!$ has a right adjoint, denoted $Rf^!:D(\Shv(Y))\to D(\Shv(X))$. This is called the exceptional inverse image.

We are now ready to define perverse sheaves.

Definition: Let $A^\bullet \in D(\Shv(X))$. Then:

• the $i$th cohomology sheaf of $A^\bullet$ is $H^i(A^\bullet) = \ker(d^i)/\im(d^i)$,
• $A^\bullet$ is a constructible complex if $H^i(A^\bullet)$ is a constructible sheaf for all $i$,
• $A^\bullet$ is a perverse sheaf if $A^\bullet\in D_b(\Shv(X))$ is constructible and $\dim(\supp(H^{-i}(P))) \leqslant i$ for all $i\in \Z$ and for $P=A^\bullet$ and $P=(A^\bullet)^\vee = (A^\vee)^\bullet$ the dual complex of sheaves.

We finish off with an example.

Example: Let $X = \R$ be a stratified space, with $X_0=0$ the origin and $X_1 = \R\setminus 0$. Let $\mathcal F\in \Shv(X)$ be an $\R$-valued sheaf given by $\mathcal F(U) = \inf_{x\in U} |x|$, and define a chain complex $A^\bullet$ in the following way:
\[0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.\]
Note that for any $U\subseteq \R$, we have $H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset$ if $0\not\in U$, and $0$ otherwise. Hence $\supp(H^{-1}(A^\bullet)) = \R\setminus 0$, whose dimension is 1. Next, $H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0$, and so $\dim(\supp(H^0(A^\bullet))) = 0$. Note that $A^\bullet$ is self-dual and constructible, as the cohomology sheaves are locally constant. Hence $A^\bullet$ is a perverse sheaf.

References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")

The Grassmannian is a complex manifold

 Lecture topic

Let $Gr(k,\C^n)$ be the space of $k$-dimensional complex subspaces of $\C^n$, also known as the complex Grassmannian. We will show that it is a complex manifold of dimension $k(n-k)$. Thanks to Jinhua Xu and professor Mihai Păun for explaining the details.

To begin, take $P\in Gr(k,\C^n)$ and an $n-k$ subspace $Q$ of $\C^n$, such that $P\cap Q = \{0\}$. Then $P\oplus Q = \C^n$, so we have natural projections

A neighborhood of $P$, depending on $Q$ may be described as $U_Q = \{S\in Gr(k,\C^n)\ :\ S\cap Q = \{0\}\}$. We claim that $U_Q \cong \Hom(P,Q)$. The isomorphism is described by
\[
\begin{array}{r c c c l}
\Hom(P,Q) & \to & U_Q & \to & \Hom(P,Q), \\
A & \mapsto & \{v+Av\ :\ v\in P\}, \\
& & S & \mapsto & \left(\pi_Q|_S\right) \circ \left(\pi_P|_S\right)^{-1}.
\end{array}
\]
The map on the right, call it $\varphi_Q$, is also the chart for the manifold structure. The idea of decomposing $\C^n$ into $P$ and $Q$ and constructing a homomorphism from $P$ to $Q$ may be visualized in the following diagram.

Then $\Hom(P,Q) \cong \Hom(\C^k,\C^{n-k})\cong \C^{k(n-k)}$, so $Gr(k,\C^n)$ is locally of complex dimension $k(n-k)$. To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let $P,P'\in Gr(k,\C^n)$ and $Q,Q'\in Gr(n-k,\C^n)$ such that $P\cap Q = P'\cap Q' = \{0\}$. Let $X\in \Hom(P,Q)$ such that $X\in \varphi_Q(U_Q\cap U_{Q'})$, with $\varphi_Q(S)=X$ and $\varphi_{Q'}(S)=X'$ for some $S\in U_Q\cap U_{Q'}$. Define $I_X(v) = v+Xv$, and note the transition map takes $X$ to
\begin{align*}
X' & = \varphi_{Q'}\circ \varphi_Q^{-1}(X) & (\text{definition}) \\
& = \varphi_{Q'}(S) & (\text{assumption})\\
& = \left(\pi_{Q'}|_S\right)\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{definition}) \\
& = \left(\pi_{Q'}|_S\right)\circ I_X\circ I_X^{-1}\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{creative identity}) \\
& = \left(\pi_{Q'}|_S\circ I_X\right)\circ \left(\pi_{P'}|_S\circ I_X\right)^{-1} & (\text{redistribution}) \\
& = \left(\pi_{Q'}|_P +\pi_{Q'}|_Q\circ X\right) \circ \left(\pi_{P'}|_P +\pi_{P'}|_Q\circ X\right). & (\text{definition})
\end{align*}
At this last step we have compositions and sums of homomorphisms and linear maps, which are all holomorphic. Hence the transition functions of $Gr(k,\C^n)$ are holomorphic, so it is a complex manifold.

The real and complex Jacobian

 Lecture topic

Let $f:\C^n\to \C^n$ be a holomorphic function. We will show that the the real Jacobian is the square of the complex Jacobian. Write $f = (f_1,\dots,f_n)$ with $f_i = u_i+\img v_i$, where the $u_i$ are functions of the $z_j = x_j+\img y_j$. By the Cauchy-Riemann equations
\[
\frac{\dy u_i}{\dy x_j} = \frac{\dy v_i}{\dy y_j}
\hspace{1cm}
\text{and}
\hspace{1cm}
\frac{\dy u_i}{\dy y_j} = -\frac{\dy v_i}{\dy x_j}
\]
and expanding, we have that
\begin{align*}
\frac{\dy f_i}{\dy z_j} & = \frac 12\left(\frac{\dy f_i}{\dy x_j} - \img \frac{\dy f_i}{\dy y_j}\right) \\
& = \frac12\left(\frac{\dy u_i}{\dy x_j} + \img \frac{\dy v_i}{\dy x_j} - \img\left(\frac{\dy u_i}{\dy y_j} + \img \frac{\dy v_i}{\dy y_j}\right)\right) \\
& = \frac12 \left(\frac{\dy u_i}{\dy x_j}+ \frac{\dy v_i}{\dy y_j} + \img \left(\frac{\dy v_i}{\dy x_j} - \frac{\dy u_i}{\dy y_j}\right)\right) \\
& = \frac{\dy u_i}{\dy x_j} + \img \frac{\dy v_i}{\dy x_j}.
\end{align*}
The complex Jacobian of $f$ is $J_\C f$ (or its determinant), with entries
\[
(J_\C f)_{i,j} = \frac{\dy f_i}{\dy z_j},
\]
and the real Jacobian of $f$ is $J_\R f$ (or its determinant), with entries
\begin{align*}
\begin{bmatrix}
(J_\R f)_{2i-1,2j-1} & (J_\R f)_{2i-1,2j} \\ (J_\R f)_{2i,2j-1} & (J_\R f)_{2i,2j}
\end{bmatrix}
& = \begin{bmatrix}
\displaystyle \frac{\dy u_i}{\dy x_j} & \displaystyle \frac{\dy u_i}{\dy y_j} \\
\displaystyle \frac{\dy v_i}{\dy x_j} & \displaystyle \frac{\dy v_i}{\dy y_j}
\end{bmatrix} \\
& \tov{R_{2i-1} + \img R_{2i} \to R_{2i-1}}
\begin{bmatrix}
\displaystyle \frac{\dy f_i}{\dy z_j} & \displaystyle \img\frac{\dy f}{\dy z} \\
\displaystyle \frac{\dy v_i}{\dy x_j} & \displaystyle \frac{\dy v_i}{\dy y_j}
\end{bmatrix} \\
& \tov{C_{2j} - \img C_{2j-i} \to C_{2j}}
\begin{bmatrix}
\displaystyle \frac{\dy f_i}{\dy z_j} & 0 \\
\displaystyle \frac{\dy v_i}{\dy x_j} & \displaystyle \overline{\frac{\dy f_i}{\dy z_j}}
\end{bmatrix},
\end{align*}
where the row and column operations have been performed for all rows $2i$ and all columns $2j$. Moving all the odd-indexed columns to the left and all odd-indexed rows to the top, we get that
\[
J_\R f \simeq \begin{bmatrix}
A & 0 \\ * & B
\end{bmatrix}
\hspace{1cm}
\text{with}
\hspace{1cm}
A_{i,j} = \frac{\dy f_i}{\dy z_j},\ \ \
B_{i,j} = \overline{\frac{\dy f_i}{\dy z_j}}.
\]
Since the number of operations to switch the columns is the same as the number of operations to switch the rows, the sign of the determinant of $J_\R f$ will not change. That is,
\[
\det(J_\R f) = \det(A)\det(B) = \det(J_\C f) \overline{\det( J_\C f)} = |\det(J_\C f)|^2.
\]

Smooth projective varieties as Kähler manifolds

Definition: Let $k$ be a field and $\P^n$ projective $n$-space over $k$. An algebraic variety $X\subset \P^n$ is the zero locus of a collection of homogeneous polynomials $f_i\in k[x_0,\dots,x_n]$.

Here we let $k=\C$, the complex numbers. Complex projective space $\C\P^n$ may be described as a complex manifold, with open sets $U_i = \{(x_0:\cdots:x_n)\ :\ x_i\neq 0\}$ and maps
\[
\begin{array}{r c l}
\varphi_i\ :\ U_i & \to & \C^n, \\
(x_0:\cdots:x_n) & \mapsto & \left(\frac{x_0}{x_i},\dots,\widehat{\frac{x_i}{x_i}},\dots,\frac{x_n}{x_i}\right),
\end{array}
\]
which can be quickly checked to agree on overlaps. In this context we assume all varieties are smooth, so they are submanifolds of $\C\P^n$.

Definition: An almost complex manifold is a real manifold $M$ together with a vector bundle endomorphism $J:TM\to TM$ (called a complex structure) with $J^2=-\id$.

Note that every complex manifold admits an almost complex structure on its underlying real manifold. Indeed, given standard coordinates $z_i=x_i+y_i$ for $i=1,\dots,n$ on $\C^n$, we get a basis $\partial/\partial x_1, \dots, \partial /\partial x_n$, $\partial/\partial y_1, \dots, \partial/\partial y_n$ on the underlying real tangent space $T_pU$, for $p\in M$ and $U\owns p$ a neighborhood. Then $J$ is defined by
\[
J\left(\frac\partial{\partial x_i}\right) = \frac\partial{\partial y_i}
\hspace{1cm},\hspace{1cm}
J\left(\frac\partial{\partial y_i}\right) = -\frac\partial{\partial x_i}.
\]
Write $T_\C M=TM\otimes_\R\C$ for the complexification of the tangent bundle, which admits a canonical decomposition $T_\C M = T^{1,0}M\oplus T^{0,1}M$, where $J|_{T^{1,0}}=i\cdot \id$ and $J|_{T^{0,1}}=(-i)\cdot \id$. We call $T^{1,0}M$ the holomorphic tangent bundle of $M$ and $T^{0,1}M$ the antiholomorphic tangent bundle of $M$, even though it is extraneous to consider any related map here as holomorphic. Define vector bundles (or sheaves, to consider sections on open sets)
\[
A^k_M = \textstyle \bigwedge^k(T_\C M)^*,
\hspace{1cm}
A^{p,q}_M = \textstyle \bigwedge^p(T^{1,0}M)^* \otimes_\C \bigwedge^q(T^{0,1}M)^*,
\]
where we drop the subscript $M$ when the context makes it clear. There is a canonical decomposition $A^k = \bigoplus_{p+q=k} A^{p,q}$, which yields projection maps $\pi^{p,q}:A^k \to A^{p,q}$. The exterior differential $d$ on $T^*M$ may be extended $\C$-linearly to $(T_\C M)^*$, and hence also to $A^k$. Define two new maps
\begin{align*}
\partial = \pi^{p+1,q}\circ d|_{A^{p,q}}\ :\ &\ A^{p,q} \to A^{p+1,q}, \\
\bar\partial = \pi^{p,q+1}\circ d|_{A^{p,q}}\ :\ &\ A^{p,q} \to A^{p,q+1}.
\end{align*}
These satisfy the Leibniz rule and (under mild assumptions) $\partial^2 = \bar\partial^2 = 0$ and $\partial \bar \partial = -\bar \partial \partial$.

From now on, the manifold $M$ will be complex with the natural complex structure described above.

Definition: A Riemannian metric on $M$ is a function $g:TM\times TM \to C^\infty(M)$ such that for all $V,W\in TM$,

• $g(V,W)=g(W,V)$, and
• $g_p(V_p,V_p)\geqslant 0$ for all $p\in M$, with equality iff $V=0$.

A Riemannian manifold is a pair $(M,g)$ where $g$ is Riemannian.

Locally we write $g_p:T_pM\times T_pM \to \R$, defined as $g_p(V_p,W_p)=g(V,W)(p)$. If $x_1,\dots,x_n$ are local coordinates on some open set $U\subset M$, then $g=\sum_{i,j}g_{ij}dx_i\wedge dx_j\in A^2(M)$, for $g_{ij} = g(\frac\partial{\partial x_i},\frac \partial{\partial x_j})\in C^\infty(U)$. Writing $V = \sum_if_i\frac\partial{\partial x_i}$ and $W=\sum_jg_j\frac\partial{\partial x_j}$, we get the local expression
\[
g_p(V_p,W_p) = \sum_{i,j}g_{ij}(p)f_i(p)g_j(p).
\]

Definition: A Hermitian metric on a complex manifold $M$ is a Riemannian metric $g$ such that $g(JV,JW)=g(V,W)$ for all $V,W\in TM$. A Hermitian manifold is a pair $(M,g)$ where $g$ is Hermitian.

There is an induced form $\omega:TM \times TM\to C^\infty(M)$ given by $\omega (V,W)=g(JV,W)$, called the fundamental form. From $g$ being Hermitian it follows that $\omega\in A^{1,1}(M)\subset A^2(M)$. Note also that any two of the structures $J,g,\omega$ determine the remaining one.

Definition: A Kähler metric on a complex manifold $M$ is a Hermitian metric whose fundamental form is closed (that is, $d\omega = 0$). A Kähler manifold is a pair $(M,g)$ where $g$ is Kähler.

Example: Recall the atlas given to $\C\P^n$ above. There is a metric (canonical in some sense) on each $U_j$ given by
\[
\omega_j = \frac i{2\pi} (\partial \circ \bar\partial) \left(\log\left(\sum_{\ell=0}^n \left|\frac{x_\ell}{x_j}\right|^2 \right)\right),
\]
called the Fubini--Study metric. Each $\omega_j$ is a section of $A^{1,1}(U_j)$, and as a quick calculation shows that $\omega_j|_{U_j\cap U_k} = \omega_k|_{U_j\cap U_k}$, there is a global metric $\omega_{FS}\in A^{1,1}(\C\P^n)$ such that $\omega_{FS}|_{U_j} = \omega_j$ for all $j$.

Hence $\C\P^n$ is a Kähler manifold. If we have a smooth projective variety $X\subset \C\P^n$, then it is a submanifold of $\C\P^n$, so by restricting $\omega_{FS}$ to $X$, we get that $X$ is also a Kähler manifold. Therefore all smooth projective varieties are Kähler.

References: Huybrechts (Complex Geometry, Chapters 1.3, 2.6, 3.1), Lee (Riemannian manifolds, Chapter 3)