# Chern classes as degeneracy loci of generic sections

We specialise slightly and consider ${\displaystyle p:E\mapsto M}$, a complex vector bundle of rank ${\displaystyle r}$, with base space ${\displaystyle M}$ a compact (orientable) smooth manifold of real dimension ${\displaystyle n}$.

A generic section is a section of the bundle which intersects the zero-section ${\displaystyle {\mathcal {O}}_{M}}$ transversally. Two sections are said to have transversal intersection if at every point of intersection, the tangent spaces of the sections at the point generate the tangent space of the ambient space at the point.

We can define the Chern classes in terms of the zero sets of generic sections, namely the transversal intersections of the generic sections with ${\displaystyle {\mathcal {O}}_{M}}$.

For a generic section ${\displaystyle s}$, we write ${\displaystyle Z(s)}$ for the zero-set of ${\displaystyle s}$. Since ${\displaystyle s}$ is transverse to the zero section, ${\displaystyle Z(s)}$ is a submanifold of ${\displaystyle M}$ of real codimension ${\displaystyle 2r}$. Applying Poincare' duality to the fundamental class ${\displaystyle [Z(s)]\in H_{n-2r}(M)}$, we obtain a cohmology class in ${\displaystyle H^{2r}(M)}$. This is called the Euler class, ${\displaystyle e(E)}$ of the vector bundle and is also the ${\displaystyle r^{th}}$ Chern class.

${\displaystyle e(E)=c_{r}(E):=[Z(s)]\in H_{n-2r}(M)\cong H^{2r}(M;\mathbb {Z} )}$
By analogy we extend the definition to give us a sequence of cohomology classes:

${\displaystyle c_{i}(E):=PD([Z(s_{1}\wedge \ldots \wedge s_{r-i+1})])\in H^{2i}(M;\mathbb {Z} )}$
for ${\displaystyle s_{1},\ldots ,s_{r-i+1}}$ generic sections. Here ${\displaystyle Z(s_{1}\wedge \ldots \wedge s_{r-i+1})}$ can be viewed as the set of points ${\displaystyle m\in M}$ where the sections ${\displaystyle s_{1}(m),\ldots ,s_{r-i+1}(m)}$ become linearly dependent. Note then that
${\displaystyle c_{1}(E):=PD([Z(s_{1}\wedge \ldots \wedge s_{r})])H^{2r}(M;\mathbb {Z} )}$

We define ${\displaystyle c_{0}(E)=1}$ and call the sum ${\displaystyle c(E)=1+c_{1}(E)+c_{2}(E)+\ldots \in H^{*}(M;Z)}$ the total Chern class of ${\displaystyle E}$. Point (3) in Theorem 4.2 below demonstrates that such a sum is finite, i.e. a well-defined element of ${\displaystyle H^{*}(M;Z)}$

Theorem 4.2

There is a unique sequence of functions ${\displaystyle c_{1},c_{2},\ldots }$ assigning to each complex vector bundle ${\displaystyle E\rightarrow B}$ a class ${\displaystyle c_{i}(E)\in H^{2i}(B,\mathbb {Z} )}$ depending only on the isomorphism type of ${\displaystyle E}$ and satisfying:

1. ${\displaystyle c_{i}(f^{*}E)=f^{*}(c_{i}(E))}$ for a pullback ${\displaystyle f^{*}E}$
2. ${\displaystyle c(E_{1}\oplus E_{2})=c(E_{1})\cup c(E_{2})}$
3. ${\displaystyle c_{i}(E)=0}$ if ${\displaystyle i>rankE}$
4. For the canonical line bundle ${\displaystyle E\rightarrow \mathbb {C} P^{\infty }}$, ${\displaystyle c_{1}(E)}$ generates ${\displaystyle H^{2}(\mathbb {C} P^{\infty };\mathbb {Z} )}$.