We specialise slightly and consider
, a complex vector bundle of rank
, with base space
a compact (orientable) smooth manifold of real dimension
.
A generic section is a section of the bundle which intersects the zero-section
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
.
For a generic section
, we write
for the zero-set of
. Since
is transverse to the zero section,
is a submanifold of
of real codimension
. Applying Poincare' duality to the fundamental class
, we obtain a cohmology class in
. This is called the Euler class,
of the vector bundle and is also the
Chern class.
![{\displaystyle e(E)=c_{r}(E):=[Z(s)]\in H_{n-2r}(M)\cong H^{2r}(M;\mathbb {Z} )}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/c3843f178ad05e9a5728129456561d93e60ec705)
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} )}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/038615ff97807673b2e224655781436302eb89f1)
for

generic sections. Here

can be viewed as the set of points

where the sections

become linearly dependent.
Note then that
![{\displaystyle c_{1}(E):=PD([Z(s_{1}\wedge \ldots \wedge s_{r})])H^{2r}(M;\mathbb {Z} )}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/15ccd3d342369897964d0ee44663975d101cee3e)
We define
and call the sum
the total Chern class of
. Point (3) in Theorem 4.2 below demonstrates that such a sum is finite, i.e. a well-defined element of
Theorem 4.2
There is a unique sequence of functions
assigning to each complex vector bundle
a class
depending only on the isomorphism type of
and satisfying:
for a pullback 

if 
- For the canonical line bundle
,
generates
.