Chern classes as degeneracy loci of generic sections

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.

By analogy we extend the definition to give us a sequence of cohomology classes:

for generic sections. Here can be viewed as the set of points where the sections become linearly dependent. Note then that

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:

  1. for a pullback
  2. if
  3. For the canonical line bundle , generates .
 
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