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Chern class

Geometric approach[edit]

Basic idea and motivation[edit]

Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. 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, however, is not true.

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 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.

Construction[edit]

There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.

The original approach to Chern classes was via algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space such that the bundle V is equal to the pullback, by f, of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles.

It can be shown that for any two maps f, g from M to the classifying space whose pullbacks are the same bundle V, the maps must be homotopic. Therefore, the pullback by either forg of any universal Chern class to a cohomology class of M must be the same class. This shows that the Chern classes of V are well-defined.

Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the Chern–Weil theory.

There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case.

Chern classes arise naturally in algebraic geometry. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, locally free sheaves) over any nonsingular variety. Algebro-geometric Chern classes do not require the underlying field to have any special properties. In particular, the vector bundles need not necessarily be complex.

Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem). Although that is strictly speaking a question about a real vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.

See Chern–Simons theory for more discussion.

The Chern class of line bundles[edit]

(Let X be a topological space having the homotopy type of a CW complex.)

An important special case occurs when V is a line bundle. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Euler class of the bundle.

The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking. That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of ${\displaystyle H^{2}(X;\mathbb {Z} )}$, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism):

In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) holomorphic line bundlesbylinear equivalence classes of divisors.

For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.

Constructions[edit]

Via the Chern–Weil theory[edit]

Given a complex hermitian vector bundle Vofcomplex rank n over a smooth manifold M, representatives of each Chern class (also called a Chern form) ${\displaystyle c_{k}($V)} ofV are given as the coefficients of the characteristic polynomial of the curvature form ${\displaystyle \Omega }$ofV.

The determinant is over the ring of ${\displaystyle n\times n}$ matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form ${\displaystyle \Omega }$ofV is defined as

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 classes in the sense of de Rham cohomology. It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in V.

If follows from the matrix identity ${\displaystyle \mathrm {tr} (\ln($X))=\ln(\det(X))} that ${\displaystyle \det($X)=\exp(\mathrm {tr} (\ln(X)))} . Now applying the Maclaurin series for ${\displaystyle \ln(X+I)}$, we get the following expression for the Chern forms:

Via an Euler class[edit]

One can define a Chern class in terms of an Euler class. This is the approach in the book by Milnor and Stasheff, and emphasizes the role of an orientation of a vector bundle.

The basic observation is that a complex vector bundle comes with a canonical orientation, ultimately because ${\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}$ is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class (the Euler class of the underlying real vector bundle) and handles lower Chern classes in an inductive fashion.

The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let ${\displaystyle \pi \colon E\to B}$ be a complex vector bundle over a paracompact space B. Thinking of B as being embedded in E as the zero section, let ${\displaystyle B'=E\setminus B}$ and define the new vector bundle:

${\displaystyle E'}$

${\displaystyle \pi |_{B'}\colon B'\to B}$

${\displaystyle \pi |_{B'}^{*}}$

${\displaystyle k<2n-1}$

It then takes some work to check the axioms of Chern classes are satisfied for this definition.

See also: The Thom isomorphism.

Examples[edit]

The complex tangent bundle of the Riemann sphere[edit]

Let ${\displaystyle \mathbb {CP} ^{1}}$ be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann sphere. Let ${\displaystyle V=T\mathbb {CP} ^{1}}$ be the bundle of complex tangent vectors having the form ${\displaystyle a\partial /\partial z}$ 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.,

This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that

Consider the Kähler metric

One readily shows that the curvature 2-form is given by

Furthermore, by the definition of the first Chern class

We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere:

This proves that ${\displaystyle T\mathbb {CP} ^{1}}$ is not a trivial vector bundle.

Complex projective space[edit]

There is an exact sequence of sheaves/bundles:^[4]

${\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}}$

${\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}($1)}

There are two ways to get the above sequence:

${\displaystyle \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}}$

${\displaystyle {\mathcal {O}}_{U}}$

${\displaystyle d(z_{i}/z_{0})}$

${\displaystyle e_{i}}$

Let L be a line in ${\displaystyle \mathbb {C} ^{n+1}}$ that passes through the origin. It is an elementary geometry to see that the complex tangent space to ${\displaystyle \mathbb {C} \mathbb {P} ^{n}}$ at the point L is naturally the set of linear maps from L to its complement. Thus, the tangent bundle ${\displaystyle T\mathbb {C} \mathbb {P} ^{n}}$ can be identified with the hom bundle

$${\displaystyle \operatorname {Hom} ({\mathcal {O}}(-1),\eta )}$$

where η is the vector bundle such that ${\displaystyle {\mathcal {O}}(-1)\oplus \eta ={\mathcal {O}}^{\oplus (n+1)}}$. It follows:

$${\displaystyle T\mathbb {C} \mathbb {P} ^{n}\oplus {\mathcal {O}}=\operatorname {Hom} ({\mathcal {O}}(-1),\eta )\oplus \operatorname {Hom} ({\mathcal {O}}(-1),{\mathcal {O}}(-1))={\mathcal {O}}($$1)^{\oplus (n+1)}.}

By the additivity of total Chern class ${\displaystyle c=1+c_{1}+c_{2}+\cdots }$ (i.e., the Whitney sum formula),

${\displaystyle H^{2}(\mathbb {C} \mathbb {P} ^{n},\mathbb {Z} )}$

${\displaystyle {\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(-1)}$

${\displaystyle c_{1}(E^{*})=-c_{1}($E)}

${\displaystyle E^{*}}$

In particular, for any ${\displaystyle k\geq 0}$,

Chern polynomial[edit]

A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle E, the Chern polynomial c_tofE is given by:

This is not a new invariant: the formal variable t simply keeps track of the degree of c_k(E).^[6] In particular, ${\displaystyle c_{t}($E)} is completely determined by the total Chern classofE: ${\displaystyle c($E)=1+c_{1}(E)+\cdots +c_{n}(E)} and conversely.

The Whitney sum formula, one of the axioms of Chern classes (see below), says that c_t is additive in the sense:

${\displaystyle E=L_{1}\oplus \cdots \oplus L_{n}}$

${\displaystyle a_{i}($E)=c_{1}(L_{i})}

${\displaystyle a_{i}($E)}

${\displaystyle c_{t}($E)}

Example: We have polynomials s_k

${\displaystyle s_{1}=\sigma _{1},s_{2}=\sigma _{1}^{2}-2\sigma _{2}}$

Example: The Todd classofE is given by:

Remark: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let G_n be the infinite Grassmannianofn-dimensional complex vector spaces. It is a classifying space in the sense that, given a complex vector bundle E of rank n over X, there is a continuous map

Remark: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let ${\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }}$ be the contravariant functor that, to a CW complex X, assigns the set of isomorphism classes of complex vector bundles of rank n over X and, to a map, its pullback. By definition, a characteristic class is a natural transformation from ${\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }=[-,G_{n}]}$ to the cohomology functor ${\displaystyle H^{*}(-,\mathbb {Z} ).}$ Characteristic classes form a ring because of the ring structure of cohomology ring. Yoneda's lemma says this ring of characteristic classes is exactly the cohomology ring of G_n:

Computation formulae[edit]

Let E be a vector bundle of rank r and ${\displaystyle c_{t}($E)=\sum _{i=0}^{r}c_{i}(E)t^{i}} the Chern polynomial of it.

• For the dual bundle ${\displaystyle E^{*}}$of${\displaystyle E}$, ${\displaystyle c_{i}(E^{*})=(-1)^{i}c_{i}($E)} .^[7]
• IfL is a line bundle, then^[8][9]
  $${\displaystyle c_{t}(E\otimes L)=\sum _{i=0}^{r}c_{i}($$E)c_{t}(L)^{r-i}t^{i}}
  and so ${\displaystyle c_{i}(E\otimes L),i=1,2,\dots ,r}$ are
  $${\displaystyle c_{1}($$E)+rc_{1}(L),\dots ,\sum _{j=0}^{i}{\binom {r-i+j}{j}}c_{i-j}(E)c_{1}(L)^{j},\dots ,\sum _{j=0}^{r}c_{r-j}(E)c_{1}(L)^{j}.}
  
• For the Chern roots ${\displaystyle \alpha _{1},\dots ,\alpha _{r}}$of${\displaystyle E}$,^[10]
  $${\displaystyle {\begin{aligned}c_{t}(\operatorname {Sym} ^{p}E)&=\prod _{i_{1}\leq \cdots \leq i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t),\\c_{t}(\wedge ^{p}E)&=\prod _{i_{1}<\cdots <i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t).\end{aligned}}}$$
  In particular, ${\displaystyle c_{1}(\wedge ^{r}E)=c_{1}($E).}
• For example,^[11] for ${\displaystyle c_{i}=c_{i}($E)} ,

  when ${\displaystyle r=2}$, ${\displaystyle c(\operatorname {Sym} ^{2}E)=1+3c_{1}+2c_{1}^{2}+4c_{2}+4c_{1}c_{2},}$

  when ${\displaystyle r=3}$, ${\displaystyle c(\operatorname {Sym} ^{2}E)=1+4c_{1}+5c_{1}^{2}+5c_{2}+2c_{1}^{3}+11c_{1}c_{2}+7c_{3}.}$

(cf. Segre class#Example 2.)

Applications of formulae[edit]

We can use these abstract properties to compute the rest of the chern classes of line bundles on ${\displaystyle \mathbb {CP} ^{1}}$. Recall that ${\displaystyle {\mathcal {O}}(-1)^{*}\cong {\mathcal {O}}($1)} showing ${\displaystyle c_{1}({\mathcal {O}}($1))=1\in H^{2}(\mathbb {CP} ^{1};\mathbb {Z} )} . Then using tensor powers, we can relate them to the chern classes of ${\displaystyle c_{1}({\mathcal {O}}($n))=n} for any integer.

Properties[edit]

Given a complex vector bundle E over a topological space X, the Chern classes of E are a sequence of elements of the cohomologyofX. The k-th Chern classofE, which is usually denoted c_k(E), is an element of

Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.^{[clarification needed]}

Classical axiomatic definition[edit]

The Chern classes satisfy the following four axioms:

1. ${\displaystyle c_{0}($E)=1} for all E.
2. Naturality: If ${\displaystyle f:Y\to X}$iscontinuous and f*E is the vector bundle pullbackofE, then ${\displaystyle c_{k}(f^{*}E)=f^{*}c_{k}($E)} .
3. Whitney sum formula: If ${\displaystyle F\to X}$ is another complex vector bundle, then the Chern classes of the direct sum ${\displaystyle E\oplus F}$ are given by
   $${\displaystyle c(E\oplus F)=c($$E)\smile c(F);}
   that is,
   $${\displaystyle c_{k}(E\oplus F)=\sum _{i=0}^{k}c_{i}($$E)\smile c_{k-i}(F).}
   
4. Normalization: The total Chern class of the tautological line bundle over ${\displaystyle \mathbb {CP} ^{k}}$ is 1−H, where HisPoincaré dual to the hyperplane ${\displaystyle \mathbb {CP} ^{k-1}\subseteq \mathbb {CP} ^{k}}$.

Grothendieck axiomatic approach[edit]

Alternatively, Alexander Grothendieck (1958) replaced these with a slightly smaller set of axioms:

• Naturality: (Same as above)
• Additivity: If ${\displaystyle 0\to E'\to E\to E''\to 0}$ is an exact sequence of vector bundles, then ${\displaystyle c($E)=c(E')\smile c(E'')} .
• Normalization: If E is a line bundle, then ${\displaystyle c($E)=1+e(E_{\mathbb {R} })} where ${\displaystyle e(E_{\mathbb {R} })}$ is the Euler class of the underlying real vector bundle.