# Friday, November 05

> References: Chriss-Ginzburg (for an introduction), Fulton's *Intersect Theory* (does a lot).

:::{.remark}
Today: Borel-Moore homology.
For example, characteristic cycles of $D\dash$modules live here.
Useful because e.g. $H_*(\CC; \ZZ) = \CC[0]$, which doesn't see that $\dim_\RR \CC = 2$.
On the other hand, $\bar{H}_*(\CC; \ZZ) = \CC[2]$, where $\bar{H}_*$ denotes taking Borel-Moore homology.
It turns out that if $X$ is compact, then $\bar{H}_* \cong H_*$.
:::

:::{.definition title="?"}
If $X \embeds G/P$ be a closed embedding, or more generally $X\embeds M$ for $M$ any smooth complex manifold (or quasiprojective variety?) with $\dim_\CC M = n$, define
\[
\bar{H}_k(X) \da H^{2n-k}(G/P, (G/P)\sm X)
.\]
:::

:::{.remark}
Goal: show this homology contains certain fundamental classes in top degree $[X] \in \bar{H}_{2n} (X)$.
:::

:::{.proposition title="?"}
There is a group morphism, the **cycle class map**,
\[
\cl: A_*(X) \to\bar{H}_*(X)
,\]
such that

- $\cl$ is compatible with proper pushforward, i.e. covariant with respect to proper morphisms.
  When $X \mapsvia{f} Y$ is proper, consider the pushforwards $f_*: A_*(X) \to A_*(Y)$ and $f_*' \bar{H}_*(X) \to \bar{H}_*(Y)$.
  For $Z \subseteq X$, we can write
  \[
  f_*[Z] = 
  \begin{cases}
  d [f(Z)] & \ro{f}{Z} \text{ degree } d 
  \\
  0 & \text{else}.
  \end{cases}
  .\]

- $\cl$ is compatible with Chern classes of vector bundles.

:::

:::{.remark}
Fulton sets up $A_*$ to mimic Borel-Moore homology.
:::

:::{.lemma title="Existence of fundamental classes"}
If $\dim_\CC(X) = n$ then $\bar{H}_{>2n}(X) = 0$ and $\bar{H}_{2n}(X; \ZZ)$ is a free abelian group with one generator for each irreducible component of $X$.
:::

:::{.remark}
On restrictions to opens: Let $U \injects X$ be open with $X \embeds G/P$ closed, so that $Y\da X\sm U \embeds X$ is closed.
Then $U \subset (G/P)\sm Y = (G/P) \sm (X\sm U) \subseteq G/P$ is open.
A mnemonic:

\begin{tikzcd}
	&& {G/P} \\
	\\
	X &&&& {(G/P)\sm Y} \\
	\\
	&& U
	\arrow["{\text{closed}}", from=3-1, to=1-3]
	\arrow["{\text{open}}", from=5-3, to=3-1]
	\arrow["{\text{closed}}"', from=5-3, to=3-5]
	\arrow["{\text{open}}"', from=3-5, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJYIl0sWzIsNCwiVSJdLFs0LDIsIihHL1ApXFxzbSBZIl0sWzIsMCwiRy9QIl0sWzAsMywiXFx0ZXh0e2Nsb3NlZH0iXSxbMSwwLCJcXHRleHR7b3Blbn0iXSxbMSwyLCJcXHRleHR7Y2xvc2VkfSIsMl0sWzIsMywiXFx0ZXh0e29wZW59IiwyXV0=)


Then
\[
\qty{ (G/P)\sm Y, (G/P\sm Y)\sm U } \subseteq \qty{ G/P, (G/P)\sm X} 
,\]
which yields a map
\[
\bar{H}_k(X) = H^{2n-k}(G/P, (G/P)\sm X) \to
H^{2n-k}((G/P)\sm Y, (G/P \sm Y) \sm U) = \bar{H}_k(U)
.\]

using that a subvariety of a smooth variety is again smooth of the same dimension.
So we have a map $\bar{H}_k(X) \to \bar{H}_k(U)$,
and this yields a LES:
:::

:::{.proposition title="LES in Borel-Moore homology"}
For $U \subseteq X$ closed with $X \subseteq G/P$ and $Y\da X\sm U$, there is a LES corresponding to
\[
(G/P) \sm X \subseteq (G/P) \sm Y \subset G/P 
,\]
given by

\begin{tikzcd}
	{H^{k+1}(G/P, (G/P)\sm Y)} && \cdots \\
	\\
	{H^k(G/P, (G/P)\sm Y)} && {H^k(G/P, (G/P)\sm X)} && {H^k((G/P)\sm Y, (G/P)\sm X)}
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=1-1]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwyLCJIXmsoRy9QLCAoRy9QKVxcc20gWSkiXSxbMiwyLCJIXmsoRy9QLCAoRy9QKVxcc20gWCkiXSxbNCwyLCJIXmsoKEcvUClcXHNtIFksIChHL1ApXFxzbSBYKSJdLFswLDAsIkhee2srMX0oRy9QLCAoRy9QKVxcc20gWSkiXSxbMiwwLCJcXGNkb3RzIl0sWzAsMV0sWzEsMl0sWzIsM10sWzMsNF1d)

For $\bar{H}$, this corresponds to

\begin{tikzcd}
	{\bar{H}_{2n-k}(Y)} && {\bar{H}_{2n-k}(X)} && {\bar{H}_{2n-k}(U)} \\
	\\
	{\bar{H}_{2n-k-1}(Y)} && \cdots
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXGJhcntIfV97Mm4ta30oWSkiXSxbMiwwLCJcXGJhcntIfV97Mm4ta30oWCkiXSxbNCwwLCJcXGJhcntIfV97Mm4ta30oVSkiXSxbMCwyLCJcXGJhcntIfV97Mm4tay0xfShZKSJdLFsyLDIsIlxcY2RvdHMiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbMyw0XV0=)

:::

:::{.remark}
Recall that for $B \subseteq A \subseteq X$, we got an inclusion of pairs \( (A, B) \subseteq (X, B) \subseteq (X, A) \).
Also note that we used 
\[
(G/P) \sm X = ( (G/P) \sm Y)\sm U
\]
where $Y \da X\sm U$.
:::

:::{.proof title="of lemma, there exist fundamental classes"}
Let $Y$ be the singular locus of $X$ with $n\da \dim_\CC X$, then

- $Y \subseteq X$ is closed, and
- $\dim_\CC Y < \dim_\CC X$ is strictly smaller.

Strategy: induct on $\dim X$ and use the LES applied to $U \da X\sm Y$ and $Y$.
Note that $U$ is smooth.
We have
\[
\bar{H}_{2n}(Y) \to \bar{H}_{2n}(X) \to \bar{H}_{2n}(U) \mapsvia{\delta} \bar{H}_{2n-1}(Y)
,\]
and $\bar{H}_{2n}(Y) = 0$ since $\dim Y < 2n$ and $\bar{H}_{2n-1}(Y) = 0$ for the same reason, making the middle map an isomorphism.
Write $U = \Disjoint_{0\leq i\leq \ell} U_i$ as a union of irreducible (so connected) components.
Then 
\[
\bar{H}_{2n}(U) = H^{2n-2n}(U, U \sm U) = H^0(U) = \ZZ \sumpower{\ell}
\]
where we can choose to embed $U \embeds U$ into itself since $U$ is smooth.
Any Zariski open has to intersect every irreducible component, so each such component yields a fundamental class.
:::