# Chern Classes and Intersection Theory (Wednesday, November 03)

## Chern Classes

> See Eisenbud and Harris, 3264 and All That, and Fulton.

:::{.theorem title="Klein's Transversality Theorem"}
Let $G \in \Alg\Grp$ act transitively on $X$ over $k = \kbar$ with $\ch k = 0$ and let $Y \leq X$ be a subvariety.

a. 
If $Z\leq X$ is a subvariety then there is an open dense subset of group elements $U \subseteq G$ such that $gZ \transverse Y$ generically.

b.
If $\phi: Z\to X$ is a morphism of varieties, then for a generic $g\in G$, the preimage $\phi\inv(gY)$ is generally reduced and is of the same codimension as $Y$.

c.
If $G$ is affine then $[gY] = [Y]$ in the Chow group $A(X)$.

:::

:::{.remark}
See ELC article, a consequence is Bertini's theorem.
:::

:::{.lemma title="?"}
Suppose $\mce \in \VectBundlerk{r}\slice {Y}$ and let $1\leq i\leq r$.
Let $\sigma_0, \cdots, \sigma_{r-i}$ be global sections of $\mce$ and $Y_{\sigma} = Y( \sigma_0 \wedgeprod \cdots \wedgeprod \sigma_{r-i})$ be the degeneracy locus where they are linearly dependent, so \( \sigma_0 \wedgeprod \cdots \wedgeprod \sigma_{r-i} \) are sections of $\Extalg^{r-i+1} \mce \to Y$.
Then

a. 
No component of $Y_{ \sigma}$ has codimension greater than $i$,

b. 
If the \( \sigma_i \) are general elements of $\mods{\CC}$ and \( V \subseteq H^0(\mce) \) be a subset of global sections generating $\mce$, then $Y_{ \sigma}$ is generically reduced with codimension $i$ in $Y$.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-11-03_14-06.pdf_tex} };
\end{tikzpicture}

:::

:::{.remark}
For affine $Y$, locally thinking of functions, either $f$ hits or misses completely any given irreducible component.
:::

:::{.proof title="of b"}
Let $V\in\mods{\CC}$ with $\dim_\CC V = m$ be the module of global sections \( \sigma_0, \cdots, \sigma_{m-1} \) generating $\mce$, and let $\phi:Y\to \Gr_{m-r}(V)$ be given by $y \mapsto \ker(V\to \mce_y)$.
If $W \leq V$ is a submodule of dimension $r-i+1$ spanned by \( \sigma_0, \cdots, \sigma_{r-i} \), then the locus $Y_{ \sigma} \subseteq Y$ is the preimage $\phi\inv (X_{\lambda}(W))$.
:::

:::{.remark}
We can write $X = \Gr_{m-r}(V) = G/P$ for $G = \GL(V)$ to realize it as a projective homogeneous variety.
Then $X_{\lambda}(W) = \ts{E \in \Gr_{m-r}(V) \st \dim(W \intersect E ) \geq 1}$ is a Schubert variety for any subspace $0 \leq W \leq V$.
In Young diagrams for a partition \( \lambda \), this condition corresponds to a valley:

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures/2021-11-03_14-19.xoj -->

![](figures/2021-11-03_14-22-15.png)

:::

:::{.theorem title="?"}
There is a unique way of assigning to each vector bundle $\mce$ on a $X$ (assumed smooth) a class $c(\mce) = 1 + c_1(\mce) + c_2(\mce) + \cdots \in A(X)$, noting that smooth $X$ guarantees a ring structure on the Chow group.
These satisfy

a. (Line bundles):
If $\mcl\to X$ is a line bundle then $c(\mcl) = 1 + c_1(\mcl)$ where $c_1(\mcl) \in A^1(X)$ is the class of the divisor of zeros minus the divisor of poles of any rational section of $\mcl$, defined up to rational equivalence in $A(X)$.

b. (Degeneracy locus):
If \( \sigma_0, \cdots, \sigma_{r-i} \) are global sections of $\mce$ and the degeneracy/dependence locus $Y_{ \sigma} \subseteq X$ has codimension $i$, then $c_i(\mce) = [X_{ \sigma} ] \in A^i(X)$

c. (Whitney's formula):
If $0 \to \mce \to \mcf \to \mcg \to 0$ is a SES in $\VectBundle\slice X$, then $c(\mcf) = c(\mce)c(\mcg) \in A(X)$.

d. (Functoriality/compatibility with pullback):
If $\phi:X\to Y$ then $\phi^*(c(\mce)) = c(\phi^*(\mce))$.

:::

:::{.remark}
This induces a map $c: \K(X) \to A(X)$.
Note that you can compose this with the cycle class map $A(X)\to H^*_{\sing}(X)$.
:::

## Singular Cohomology

> See Anderson-Fulton.

:::{.remark}
We can define a total Chern class $c(\mce) = \sum_i c_i(\mce) u^i \in R[u]$ for $R\da H^*_{\sing}(X)$.
:::

:::{.proposition title="?"}
Setup: take $X$ paracompact and Hausdorff/T2, which will be necessary for partitions of unity.
For $\mce \mapsvia{\pi} X \in \VectBundle(\CC)\slice X$, there exist $c_i(\mce)\in H^{2i}(X)$ satisfying

1. If $f:X\to Y \in \Top$ then $f^*(c_i(\mce)) = c_i(f^* \mce)$.

2. $c_i(\mce) = 0$ unless $o\leq i \leq r\da \rank(\mce)$, and $c_0(\mce) = 1$

3. Exact sequences of vector bundles yield Whitney's formula.

If additionally $X$ is smooth,

4. If $\mcl, \mcm$ are line bundles, then $c_1(\mcl \tensor \mcm) = c_1(\mcl) + c_1(\mcm)$.

5. If $s:X\to \mcl$ is a nonzero section, writing $Z(s) \subseteq X$ for its zero set, $[Z(s)] = c_1(\mcl) \in H^2(X)$.

6. For the projectivization $\pi: \PP(\mce)\to X$, there is a Poincaré duality: considering $\OO(-1) \subseteq \pi^* \mce$ and its dual $\OO(1)$,
\[
H^*(\PP(\mce)) = H^*(X)[\zeta] / \gens{ \zeta^r - c_1(\mce\dual)\zeta^{r-1} + \cdots + (-1)^r c_r(\mce\dual) }
.\]

:::

> This is the tautological bundle, to be continued on Friday!