Chern Class

Chern classes are named for Shiing-Shen Chern , who first gave a general definition of them in the 1940s .

BASIC IDEA AND MOTIVATION

Chern classes are characteristic classes. They are Topological Invariant s associated to vector bundles on a smooth manifold. If you describe the same vector bundle on a manifold in two different ways, the Chern classes will be the same. When are two ostensibly different vector bundles the same? When are they different? These questions can be quite hard to answer. But the Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. (The converse is not true, though.)

In topology, differential geometry, and algebraic geometry, it is often important to count how many Linearly Independent sections a vector bundle has. The Chern classes offer some information about this through, for instance, the Riemann-Roch Theorem and the Atiyah-Singer Index Theorem . Chern classes are therefore useful in modern mathematics.

Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the Curvature Form .

For these reasons, and others, Chern classes are used to attack diverse mathematical problems.

THE CHERN CLASS OF A HERMITIAN VECTOR BUNDLE ON A SMOOTH MANIFOLD

Given a complex Hermitian Vector Bundle ''V'' of Complex Rank ''n'' over a Smooth Manifold ''M'',
a representative of each Chern class (also called a Chern form)

c_k(V)

of ''V'' are given as the coefficients of the Characteristic Polynomial of the Curvature Form

\Omega

of ''V''.

:

\det \left(rac {it\Omega}{2\pi} +I
ight) = \sum_k c_k(V) t^k

The determinant is over the ring of ''n''×''n'' matrices whose entries are polynomials in ''t'' with coefficients in the commutative algebra of even complex differential forms on ''M''. The Curvature Form

\Omega

of ''V'' is defined as

:

\Omega=d\omega+rac{1}{2}

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with

\omega

the Connection Form and ''d'' the Exterior Derivative , or via the same expression in which

\omega

is a Gauge Form for the Gauge Group of ''V''. The scalar ''t'' is used here only as an Indeterminate to Generate the sum from the determinant, and ''I'' denotes the ''n''×''n'' Identity Matrix .

To say that the expression given is a ''representative'' of the Chern class indicates that 'class' here means Up To addition of an Exact Differential Form . That is, Chern classes are Cohomology Class es in the sense of De Rham Cohomology . It can be shown that the cohomology class of the Chern forms do not depend on the choice of connection in ''V''.

EXAMPLE: THE COMPLEX TANGENT BUNDLE OF THE RIEMANN SPHERE

Let CP¹ be the . Suppose that ''z'' is a Holomorphic Local Coordinate for the Riemann sphere. Let ''V'' = '''TCP'''¹ be the bundle of complex tangent vectors having the form ''a''∂/∂''z'' at each point, where ''a'' is a complex number. We prove the complex version of the '' Hairy Ball Theorem '': ''V'' has no section which is everywhere nonzero.

For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e.,

:

c_1({\mathbf C\mathbf P}^1	imes {\mathbf C})=0.

This is evinced by the fact that a trivial bundle always admits a flat metric.

So, we shall show that

:

c_1(V) 
ot= 0.

Consider the Kähler Metric

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Information About

Chern Class