Chern classes

In mathematics , more precisely in algebraic topology and in differential geometry and topology , Chern classes are a special type of characteristic classes that are assigned to complex vector bundles .

Chern classes are named after Shiing-Shen Chern , who first generally defined them in the 1940s.

Basic idea and motivation

Chern classes are characteristic classes. They are therefore topological invariants of complex vector bundles over smooth manifolds and two isomorphic vector bundles thus have the same Chern classes. The Chern classes thus provide a way to verify that two vector bundles are different over a smooth manifold, but they cannot be used to decide that two vector bundles are isomorphic (since non-isomorphic vector bundles can have the same Chern class).

In topology, differential geometry and algebraic geometry it is often important to determine the maximum number of linearly independent sections of a vector bundle. Chern classes make it possible to obtain information about this, for example with the Riemann-Roch theorem or the Atiyah-Singer index theorem . This is one of the reasons why Chern classes are an important tool in modern mathematics.

In many cases, Chern classes can also be explicitly calculated in practice. In differential geometry (and in parts of algebraic geometry) Chern classes can be expressed as polynomials in the coefficients of the curvature shape.

Therefore, Chern classes are used to tackle a wide variety of mathematical problems. They are also used in physics.

The Chern class of a Hermitian vector bundle on a smooth manifold

Let be a smooth manifold , a Hermitian vector bundle with rank over and a connection on . The -th Chern form of is then given by the coefficients of the characteristic polynomial of the curvature form of , i.e. ${\displaystyle M}$ ${\displaystyle \pi \colon V\to M}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle \nabla }$ ${\displaystyle V}$ ${\displaystyle k}$ ${\displaystyle c_{k}(\nabla )}$ ${\displaystyle V}$ ${\displaystyle \Omega }$ ${\displaystyle V}$

${\displaystyle \det \left({\frac {it\Omega }{2\pi }}+I\right)=\sum _{k}c_{k}(\nabla )t^{k}}$.

The determinant is the ring of matrices with entries from the polynomial ring over the commutative algebra of the straight complex differential forms on formed. The curvature form is Lie algebra valued, and through ${\displaystyle n\times n}$ ${\displaystyle M}$ ${\displaystyle \Omega }$

${\displaystyle \Omega =d\omega +{\frac {1}{2}}[\omega ,\omega ]}$

defined, where is the form of connection and the outer derivative . ${\displaystyle \omega }$ ${\displaystyle d}$

The -th Chern class , also denoted by, is defined as the de Rham's cohomology classes of the -th Chern form . It can be shown that the Chern class, i.e. the cohomology class of the Chern form, does not depend on the choice of the relationship in . The Chern class is thus an invariant of the vector bundle, while the Chern form depends on the chosen context. ${\displaystyle c_{k}}$ ${\displaystyle k}$ ${\displaystyle c_{k}(V)\in H_{dR}^{2k}(M;\mathbb {C} )}$ ${\displaystyle k}$ ${\displaystyle V}$

One can show that the Chern classes are in the picture of . ${\displaystyle H^{*}(M;\mathbb {Z} )\rightarrow H^{*}(M;\mathbb {C} )}$

Example: The complex tangential bundle of the Riemann sphere

Let CP ^{1 be} the Riemann number sphere , the one-dimensional complex projective space . Let us also be a holomorphic local coordinate , the complex tangential bundle, the vectors have the form at each point , denoting a complex number. We show that there is no nowhere disappearing cut . ${\displaystyle z}$ ${\displaystyle V=T\mathbf {CP} ^{1}}$ ${\displaystyle a\,\partial /\partial z}$ ${\displaystyle a}$ ${\displaystyle V}$

For this we need the following fact: The first Chern class of a trivial bundle is zero, ie

${\displaystyle c_{1}({\mathbf {C} \mathbf {P} }^{1}\times {\mathbf {C} })=0.}$

One can convince oneself of this by the fact that a trivial bundle always has a flat metric.

Now we show

${\displaystyle c_{1}(V)\not =0.}$

To do this, consider a manifold with the Kähler metric

${\displaystyle h={\frac {dzd{\bar {z}}}{(1+|z|^{2})}}.}$

The curvature shape to is then through ${\displaystyle h}$

${\displaystyle \Omega ={\frac {2dz\wedge d{\bar {z}}}{(1+|z|^{2})^{2}}}.}$

given. According to the definition of the first Chern class is

${\displaystyle c_{1}={\frac {i}{2\pi }}\Omega .}$

We have to show that the cohomology class is non-zero. To do this, we calculate the integral ${\displaystyle c_{1}}$

${\displaystyle \int c_{1}={\frac {i}{\pi }}\int {\frac {dz\wedge d{\bar {z}}}{(1+|z|^{2})^{2}}}=2}$

by coordinate transformation. According to Stokes ' theorem, on the other hand , the integral of an exact form would have the value 0, so is not trivial. ${\displaystyle T\mathbf {CP} ^{1}}$

At the same time, this example shows, by referring to Stokes' theorem, that differential topological applications of Chern's classification (e.g. in physics, see below) will primarily concern "circulation problems".

properties

Let be a complex vector bundle over topological space . The Chern classes of are a sequence of elements of the cohomology of . The -th Chern class of , which is usually denoted, is an element of , the cohomology of X (with integer coefficients). One also defines the total Chern class ${\displaystyle V}$ ${\displaystyle X}$ ${\displaystyle V}$ ${\displaystyle X}$ ${\displaystyle k}$ ${\displaystyle V}$ ${\displaystyle c_{k}(V)}$ ${\displaystyle H^{2k}(X)}$

${\displaystyle c(V)=c_{0}(V)+c_{1}(V)+c_{2}(V)+\cdots }$

as an element of . ${\displaystyle H^{*}(X)}$

The Chern classes satisfy the following four axioms:

• ${\displaystyle c_{0}(V)=1}$for each . ${\displaystyle V}$

• Functoriality : is a continuous function and the bundle retrieved by means of it is for each . ${\displaystyle f\colon Y\to X}$ ${\displaystyle f^{*}V}$ ${\displaystyle f}$ ${\displaystyle c_{k}(f^{*}V)=f^{*}c_{k}(V)}$ ${\displaystyle k}$

• Additivity: If another complex bundle is the Chern class of the Whitney sum through ${\displaystyle W\to X}$ ${\displaystyle V\oplus W}$

${\displaystyle c(V\oplus W)=c(V)\cup c(W)}$

given, that is for each is ${\displaystyle k}$

${\displaystyle c_{k}(V\oplus W)=\sum _{i=0}^{n}c_{i}(V)\cup c_{k-i}(W).}$

• Normalization: The total Chern class of the tautological bundle of lines over is . Here called the Poincaré dual of the hyperplane . ${\displaystyle \mathbf {CP} ^{k}}$ ${\displaystyle 1-H}$ ${\displaystyle H}$ ${\displaystyle \mathbf {CP} ^{k-1}\subseteq \mathbf {CP} ^{k}}$

Alexander Grothendieck replaced these axioms with somewhat weaker ones:

• Functionality: (see above)

• Additivity: If there is an exact sequence of vector bundles, then is . ${\displaystyle 0\rightarrow V'\rightarrow V\rightarrow V''\rightarrow 0}$ ${\displaystyle c(V)=c(V')\cup c(V'')}$

• Normalization: If a straight line bundle , then where denotes the Euler class of the underlying real vector bundle. ${\displaystyle V}$ ${\displaystyle c(V)=1+e(V_{\mathbf {R} })}$ ${\displaystyle e(V_{\mathbf {R} })}$ ${\displaystyle V}$

Indeed, these properties clearly characterize the Chern classes. Some important implications are

• Is the rank of , then for each (total Chern class is particularly well defined ). ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle c_{k}(V)=0}$ ${\displaystyle k>n}$

• The highest Chern class of (that is , the rank of ) is always equal to the Euler class of the underlying real vector bundle. ${\displaystyle V}$ ${\displaystyle c_{n}(V)}$ ${\displaystyle n}$ ${\displaystyle V}$

Since the Chern classes are cohomology classes with whole coefficients, they are somewhat finer than those considered in the Riemann example above, which had real coefficients.

Construction of chern classes

There are many ways of approaching the goal, each one focusing on a slightly different aspect of the Chern classes.

The original approach was algebraic topology. The infinite Graßmann manifold is the classifying space for -dimensional complex vector bundles. This means that every -dimensional complex vector bundle above the base can be obtained as a pullback of the tautological bundle under a continuous mapping . The Chern classes of the tautological bundle are called universal Chern classes and the Chern classes of a vector bundle classified by the mapping are obtained by withdrawing the universal Chern classes by means of the cohomology-induced mapping . The Chern classes of the tautological bundle can be expressed explicitly in terms of Schubert cycles. ${\displaystyle BGL(n,\mathbb {C} )=Gr(n,\infty )}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle B}$ ${\displaystyle f\colon B\to BGL(n,\mathbb {C} )}$ ${\displaystyle f}$ ${\displaystyle f^{*}}$

Chern's approach to related differential geometry and the curvature shape approach described above. He showed that both approaches are equivalent.

Alexander Grothendieck's axiomatic approach shows that the Chern classes only have to be determined for straight line bundles.

Chern classes also occur naturally in algebraic geometry . The generalized Chern classes of algebraic geometry can be defined for locally free sheaves over any non-singular variety . The Chern classes of algebraic geometry only require algebraic closure from the underlying body; in particular, vector bundles do not necessarily have to be complex.

Regardless of the approach chosen, the intuitive meaning of a Chern class is that of the 'required zeros' of a vector bundle section: For example, the statement that you cannot comb a hedgehog. Even though this is actually a question of real vector bundles (the hedgehog's "spines" are real straight lines), there are generalizations in which the spines are complex, or for the one-dimensional projective space above other bodies.

Chern classes of straight line bundles

An important special case is that of a straight line bundle . The only nontrivial Chern class in this case is the first, which is an element of the second cohomology group of . Since it is the highest Chern class, it is equal to the Euler class of the bundle. ${\displaystyle V}$ ${\displaystyle X}$

The first Chern class turns out to be a complete invariant , which classifies the complex line bundles. This means that there is a bijection between the isomorphism classes of complex straight line bundles above and the elements of ; each bundle is assigned its first Chern class. The addition in corresponds to the tensor product under this bijection . ${\displaystyle X}$ ${\displaystyle H^{2}(X)}$ ${\displaystyle H^{2}(X)}$

In algebraic geometry, this classification of the complex line bundles by the first Chern class is a first approximation of the classification of holomorphic line bundles by linear equivalence classes of divisors .

The Chern classes are no longer a complete invariant for complex bundles of a larger dimension.

Chern classes of almost complex manifolds and Cobordism theory

The theory of the Chern classes provides cobordism invariants of almost complex manifolds.

If it is an almost complex manifold, then its tangent bundle is a complex vector bundle. Its chern classes are also referred to as the chern classes of . Is compact and geradedimensional, about -dimensional, then each can monomial the total degree in the Chern classes of the fundamental class of paired and provides be an integer, a Chernzahl of . If there is another almost complex manifold of the same dimension, then and are co-ordinate if and only if they have the same Chern numbers. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle 2d}$ ${\displaystyle 2d}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

Generalizations

There is a generalization of the theory of the Chern classes in which the ordinary cohomology theory is replaced by a generalized one, which must have the additional property of complex orientability. The formal properties of the Chern classes remain the same, only as a rule, which computes the first Chern class of a tensor product of line bundles, the addition is replaced by the corresponding operation.

Chern numbers

On oriented manifolds of the dimension , every product of Chern classes can be paired with the fundamental class from the total degree and thus yields an integer, a Chern number of the vector bundle. For example, if the manifold has dimension six, the different Chern numbers result from , and . In general, the number of different Chern Numbers is the number of partitions of . ${\displaystyle 2d}$ ${\displaystyle 2d}$ ${\displaystyle c_{1}^{3}}$ ${\displaystyle c_{1}c_{2}}$ ${\displaystyle c_{3}}$ ${\displaystyle d}$

As mentioned above, the Chern numbers of the tangent bundle of an (almost) complex manifold are an important invariant.

The chern character

Chern classes can be used to construct a ring homomorphism from the topological K-theory of a space to the completion of its rational cohomology. This character is through for a bundle of straight lines ${\displaystyle V}$

${\displaystyle {\hbox{ch}}(V)=\exp(c_{1}(V))}$

given.

For sums of straight line bundles, the character is defined by additivity, which results in a representation of the character by the classes. This is used to define the character for all vector bundles that are first terms

${\displaystyle {\hbox{ch}}(V)=\dim V+c_{1}(V)+c_{1}(V)^{2}/2-c_{2}(V)+\ldots }$

If the sum of the straight line bundles with first Chern classes , then is ${\displaystyle V}$ ${\displaystyle L_{1},\ldots ,L_{k}}$ ${\displaystyle x_{1},\ldots ,x_{k}}$

${\displaystyle {\hbox{ch}}(V)=e^{x_{1}}+\dots +e^{x_{k}}.}$

In the differential geometric approach via the curvature, the Chern character is explicit

${\displaystyle {\hbox{ch}}(V)={\hbox{tr}}\left(\exp \left({\frac {i\Omega }{2\pi }}\right)\right)}$

given, denotes the curvature. ${\displaystyle \Omega }$

The Chern character helps, for example, with the calculation of the Chern classes of a tensor product. More specifically, it has the following two properties

${\displaystyle {\hbox{ch}}(V\oplus W)={\hbox{ch}}(V)+{\hbox{ch}}(W)}$

${\displaystyle {\hbox{ch}}(V\otimes W)={\hbox{ch}}(V){\hbox{ch}}(W)}$

As mentioned above, the first of these formulas can be generalized with the help of Grothendieck's axiom of additivity for Chern classes to the statement that a homomorphism of Abelian groups is from the K-theory into the rational cohomology of . The second equation says that this homomorphism is multiplicative, i.e. it is even a homomorphism of rings. ${\displaystyle {\hbox{ch}}}$ ${\displaystyle K(X)}$ ${\displaystyle X}$

Pontryagin classes

For a real vector bundle over a topological space to define the Pontryagin classes by ${\displaystyle V}$ ${\displaystyle X}$ ${\displaystyle p_{i}(V)\in H^{4i}(X)}$

${\displaystyle p_{i}(V):=c_{2i}(V\otimes {\mathbb {C} }).}$

Here is the complexification of the real vector bundle . (One can show that is always , which is why one only considers the even Chern classes.) ${\displaystyle V\otimes {\mathbb {C} }}$ ${\displaystyle V}$ ${\displaystyle c_{2i+1}(V\otimes {\mathbb {C} })=0}$

Nowikow proved that the rational Pontryagin classes of the tangent space of differentiable manifolds are invariant under homeomorphisms . But they are not invariant under homotopy equivalences . The Novikov conjecture says that (depending on the fundamental group ) certain combinations of rational Pontryagin classes are invariant under homotopy equivalences.

The signature operator stating that the signature closed differentiable manifolds as a combination of Pontrjagin classes (the L-polynomial) can be calculated. From the Atiyah-Singer index theorem, it follows that the index of the Dirac operator of a spin manifold can also be calculated as a combination of Pontryagin classes (the  polynomial).

Chern classes in algebraic geometry

Let it be a smooth projective variety over an algebraically closed body and its chow ring . Grothendieck proved that there is a clear theory of Chern classes, which assigns Chern classes to every locally free coherent sheaf such that the following axioms are fulfilled: ${\displaystyle X}$ ${\displaystyle \textstyle A(X)=\bigoplus _{r}A^{r}(X)}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle c_{i}({\mathcal {E}})\in A^{i}(X)}$

• ${\displaystyle c_{0}({\mathcal {E}})=1}$
• For each invertible sheaf is . ${\displaystyle {\mathcal {O}}(D)}$ ${\displaystyle c_{1}({\mathcal {O}}(D))=\left[D\right]}$
• For every morphism is . ${\displaystyle f\colon X\to Y}$ ${\displaystyle f^{*}(c_{i}({\mathcal {E}}))=c_{i}(f^{*}{\mathcal {E}})}$
• For every exact sequence is . ${\displaystyle 0\to {\mathcal {E}}^{\prime }\to {\mathcal {E}}\to {\mathcal {E}}^{\prime \prime }}$ ${\displaystyle c_{k}({\mathcal {E}})=\bigoplus _{i+j=k}c_{i}({\mathcal {E}}^{\prime })c_{j}({\mathcal {E}}^{\prime \prime })}$

The construction of the algebraic Chern classes is analogous to the construction of the topological Chern classes using the Leray-Hirsch theorem . In particular, for algebraic vector bundles over smooth complex projective varieties, both constructions yield the same Chern classes.

Chern classes in physics

In physics, too, the Chern classes have been increasingly used since around 2015 and are also explicitly named as such (which was not the case before), since then, not only in high-energy physics , but also increasingly in solid-state physics, new differential topological aspects are dealt with: In addition to older " Umlauf ”statements in physics, such as the Aharonov-Bohm effect of quantum mechanics or the well-known Meissner-Ochsenfeld effect of superconductivity, are now used by Chern classes in physics primarily for the differential topological classification of Umlauf singularities, especially in the so-called quantum Hall -Effect or with the so-called topological superconductors or topological insulators . The connection with mathematics arises from the fact that the magnetic flux density functions as an “effective curvature” via its vector potential , although work is carried out in a flat space. ${\displaystyle \mathbf {B} }$ ${\displaystyle \mathbf {A} }$ ${\displaystyle (\mathbf {B} ={\text{rot}}\ \mathbf {A} ),}$

More generally there is a relation to the Yang-Mills theories , whereby the mathematical term " curvature " in physics functions as " field strength ". The physical “Hilbert space state” corresponds to a projective-complex manifold, because the state should remain the same when multiplied by a complex number. ${\displaystyle |\psi \rangle }$ ${\displaystyle \psi }$

literature

• Shiing-Shen Chern : Characteristic Classes of Hermitian Manifolds. In: Annals of Mathematics. 2nd Series, 47, 1, 1946, ISSN  0003-486X , pp. 85-121.
• Alexander Grothendieck : La Théorie des classes de Chern. In: Bulletin de la Société Mathématique de France. 86, 1958, ISSN  0037-9484 , pp. 137-154, online (PDF; 1.43 MB) .
• Jürgen Jost : Riemannian Geometry and Geometric Analysis. 3rd edition. Springer-Verlag, Berlin et al. 2002, ISBN 3-540-42627-2 . ( University text ).
• John W. Milnor , James D. Stasheff : Characteristic Classes. Princeton University Press et al., Princeton NJ 1974, ISBN 0-691-08122-0 ( Annals of Mathematics Studies 76).
• Allen Hatcher: Vector Bundles and K-Theory (PDF; 1.2 MB)