Editing Chern class

{{Short description|Characteristic classes of vector bundles}}
{{Use American English|date=January 2019}}
In [[mathematics]], in particular in [[algebraic topology]], [[differential geometry and topology|differential geometry]] and [[algebraic geometry]], the '''Chern classes''' are [[characteristic class]]es associated with [[complex vector bundle|complex]] [[vector bundle]]s. They have since become fundamental concepts in many branches of mathematics and physics, such as [[string theory]], [[Chern–Simons theory]], [[knot theory]], and [[Gromov–Witten theory|Gromov–Witten invariants]].
Chern classes were introduced by {{harvs|txt|authorlink=Shiing-Shen Chern|first=Shiing-Shen|last=Chern|year=1946}}.

== Geometric approach ==

=== Basic idea and motivation ===

Chern classes are [[characteristic class]]es. They are [[topological invariant]]s 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 ===

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 cycle]]s.

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 ''f'' or ''g'' 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 (category theory)|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 ==

{{For|a sheaf theoretic description|Exponential sheaf sequence}}

(Let ''X'' be a topological space having the [[Homotopy#Homotopy equivalence|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 set of invariants|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 <math>H^2(X;\Z)</math>, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism):
<math display="block">c_1(L \otimes L') = c_1(L) + c_1(L');</math>
the [[tensor product]] of complex line bundles corresponds to the addition in the second cohomology group.<ref>{{cite book | first1=Raoul | last1=Bott| first2=Loring|last2=Tu |author1-link=Raoul Bott |title=Differential forms in algebraic topology | date=1995|publisher=Springer|location=New York [u.a.]|isbn=3-540-90613-4|page=267ff|edition=Corr. 3. print.}}</ref><ref>{{cite web |last=Hatcher |first=Allen |author-link=Allen Hatcher |title=Vector Bundles and K-theory |url=https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf |at=Proposition 3.10.}}</ref>

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 bundle]]s by [[linear equivalence]] classes of [[Divisor (algebraic geometry)|divisor]]s.

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

== Constructions ==

=== Via the Chern–Weil theory ===
{{main|Chern–Weil theory}}

Given a complex [[Hermitian metric|hermitian]] [[vector bundle]] ''V'' of [[vector bundle|complex rank]] ''n'' over a [[smooth manifold]] ''M'', representatives of each Chern class (also called a '''Chern form''') <math>c_k(V)</math> of ''V'' are given as the coefficients of the [[characteristic polynomial]] of the [[curvature form]] <math>\Omega</math> of ''V''.

<math display="block">\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k</math>

The determinant is over the ring of <math>n \times n</math> matrices whose entries are polynomials in ''t'' with coefficients in the commutative algebra of even complex differential forms on ''M''.  The [[curvature form]] <math>\Omega</math> of ''V'' is defined as
<math display="block">\Omega = d\omega+\frac{1}{2}[\omega,\omega]</math>
with ω the [[connection form]]  and ''d'' the [[exterior derivative]], or via the same expression in which ω is a [[gauge field]] for the [[gauge group]] of ''V''.  The scalar ''t'' is used here only as an [[indeterminate (variable)|indeterminate]] to [[generating function|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 classes of the Chern forms do not depend on the choice of connection in ''V''.

If follows from the  matrix identity <math>\mathrm{tr}(\ln(X))=\ln(\det(X))</math> that <math> \det(X) =\exp(\mathrm{tr}(\ln(X)))</math>. Now applying the  [[Taylor series|Maclaurin series]] for <math>\ln(X+I)</math>, we get the following expression for the Chern forms:

<math display="block">\sum_k c_k(V) t^k = \left[ 1 
       + i \frac{\mathrm{tr}(\Omega)}{2\pi} t 
       +   \frac{\mathrm{tr}(\Omega^2)-\mathrm{tr}(\Omega)^2}{8\pi^2} t^2
       + i \frac{-2\mathrm{tr}(\Omega^3)+3\mathrm{tr}(\Omega^2)\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega)^3}{48\pi^3} t^3
       + \cdots
       \right].</math>

=== Via an Euler class ===
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 <math>\operatorname{GL}_n(\Complex)</math> 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 <math>\pi\colon E \to B</math> be a complex vector bundle over a [[paracompact space]] ''B''. Thinking of ''B'' as being embedded in ''E'' as the zero section, let <math>B' = E \setminus B</math> and define the new vector bundle:
<math display="block">E' \to B'</math>
such that each fiber is the quotient of a fiber ''F'' of ''E'' by the line spanned by a nonzero vector ''v'' in ''F'' (a point of ''B′'' is specified by a fiber ''F'' of ''E'' and a nonzero vector on ''F''.)<ref>Editorial note: Our notation differs from Milnor−Stasheff, but seems more natural.</ref> Then <math>E'</math> has rank one less than that of ''E''. From the [[Gysin sequence]] for the fiber bundle <math>\pi|_{B'}\colon B' \to B</math>:
<math display="block">\cdots \to \operatorname{H}^k(B; \Z) \overset{\pi|_{B'}^*} \to \operatorname{H}^k(B'; \Z) \to \cdots,</math>
we see that <math>\pi|_{B'}^*</math> is an isomorphism for <math>k < 2n-1</math>. Let
<math display="block">c_k(E) = \begin{cases}
{\pi|_{B'}^*}^{-1} c_k(E') & k < n\\
e(E_{\R}) & k = n \\
0 & k > n
\end{cases}</math>

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

See also: [[Thom space#The Thom isomorphism|The Thom isomorphism]].<!--
== Via an elementary symmetric polynomial ==
This is the approach taken by topologists such as May or Hatcher. This approach leads very directly to related notions such as Chern characters. See the "Chern polynomial" section. -->

==Examples==

===The complex tangent bundle of the Riemann sphere===

Let <math>\mathbb{CP}^1</math> be the [[Riemann sphere]]: 1-dimensional [[complex projective space]]. Suppose that ''z'' is a [[Holomorphic function|holomorphic]] [[manifold|local coordinate]] for the Riemann sphere. Let <math>V=T\mathbb{CP}^1</math> be the bundle of complex tangent vectors having the form <math>a \partial/\partial z</math> 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.,
<math display="block">c_1(\mathbb{CP}^1\times \Complex)=0.</math>

This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that
<math display="block">c_1(V) \not= 0.</math>

Consider the [[Kähler metric]]
<math display="block">h = \frac{dz d\bar{z}}{(1+|z|^2)^2}.</math>

One readily shows that the curvature 2-form is given by
<math display="block">\Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^2)^2}.</math>

Furthermore, by the definition of the first Chern class
<math display="block">c_1= \left[\frac{i}{2\pi} \operatorname{tr} \Omega\right] .</math>

We must show that this cohomology class is non-zero.  It suffices to compute its integral over the Riemann sphere:
<math display="block">\int c_1 =\frac{i}{\pi}\int \frac{dz\wedge d\bar{z}}{(1+|z|^2)^2}=2</math>
after switching to [[polar coordinates]]. By [[Stokes' theorem]], an [[exact form]] would integrate to 0, so the cohomology class is nonzero.

This proves that <math>T\mathbb{CP}^1</math> is not a trivial vector bundle.

=== Complex projective space ===
There is an exact sequence of sheaves/bundles:<ref>The sequence is sometimes called the [[Euler sequence]].</ref>
<math display="block">0 \to \mathcal{O}_{\mathbb{CP}^n} \to \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus (n+1)} \to T\mathbb{CP}^n \to 0</math>
where <math>\mathcal{O}_{\mathbb{CP}^n} </math> is the structure sheaf (i.e., the trivial line bundle), <math>\mathcal{O}_{\mathbb{CP}^n}(1)</math> is [[Serre's twisting sheaf]] (i.e., the [[hyperplane bundle]]) and the last nonzero term is the [[tangent sheaf]]/bundle.

There are two ways to get the above sequence:

{{Ordered list
|<ref>{{harvnb|Hartshorne|loc=Ch. II. Theorem 8.13.}}</ref> Let <math>z_0, \ldots , z_n</math> be the coordinates of <math>\Complex^{n+1},</math> let <math>\pi\colon \Complex^{n+1} \setminus \{0\} \to \Complex\mathbb{P}^n</math> be the canonical projection, and let <math>U = \mathbb{CP}^n \setminus \{ z_0 = 0\}</math>. Then we have:
<math display="block">\pi^* d(z_i / z_0) = {z_0 dz_i - z_i d z_0 \over z_0^2}, \, i \ge 1.</math>
In other words, the [[cotangent sheaf]] <math>\Omega_{\Complex\mathbb{P}^n}|_U</math>, which is a free <math>\mathcal{O}_U</math>-module with basis <math>d(z_i / z_0)</math>, fits into the exact sequence
<math display="block"> 0 \to \Omega_{\Complex\mathbb{P}^n}|_U \overset{dz_i \mapsto e_i}\to \oplus_1^{n+1} \mathcal{O}(-1)|_U \overset{e_i \mapsto z_i}\to \mathcal{O}_U \to 0, \, i \ge 0,</math>
where <math>e_i</math> are the basis of the middle term. The same sequence is clearly then exact on the whole projective space and the dual of it is the aforementioned sequence.
|Let ''L'' be a line in <math>\Complex^{n+1}</math> that passes through the origin. It is an exercise in [[elementary geometry]] to see that the complex tangent space to <math>\Complex\mathbb{P}^n</math> at the point ''L'' is naturally the set of linear maps from ''L'' to its complement. Thus, the tangent bundle <math>T\Complex\mathbb{P}^n</math> can be identified with the [[hom bundle]]
<math display="block">\operatorname{Hom}(\mathcal{O}(-1), \eta)</math>
where η is the vector bundle such that <math>\mathcal{O}(-1) \oplus \eta = \mathcal{O}^{\oplus (n+1)}</math>. It follows:
<math display="block">T\Complex \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)}.</math>
}}

By the additivity of total Chern class <math>c = 1 + c_1 + c_2 + \cdots</math> (i.e., the Whitney sum formula),
<math display="block">c(\Complex\mathbb{P}^n) \overset{\mathrm{def}}= c(T\mathbb{CP}^n) = c(\mathcal{O}_{\Complex\mathbb{P}^n}(1))^{n+1} = (1+a)^{n+1},</math>
where ''a'' is the canonical generator of the cohomology group <math>H^2(\Complex\mathbb{P}^n, \Z )</math>; i.e., the negative of the first Chern class of the tautological line bundle <math>\mathcal{O}_{\Complex\mathbb{P}^n}(-1)</math> (note: <math>c_1(E^*) = -c_1(E)</math> when <math>E^*</math> is the dual of ''E''.)

In particular, for any <math>k\ge  0</math>,
<math display="block">c_k(\Complex\mathbb{P}^n) = \binom{n+1}{k} a^k.</math>

== Chern polynomial ==
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''<sub>''t''</sub> of ''E'' is given by:
<math display="block">c_t(E) =1 + c_1(E) t + \cdots + c_n(E) t^n.</math>

This is not a new invariant: the formal variable ''t'' simply keeps track of the degree of ''c''<sub>''k''</sub>(''E'').<ref>In a ring-theoretic term, there is an isomorphism of graded rings:
<math display="block">H^{2*}(M, \Z) \to \oplus_k^\infty \eta(H^{2*}(M, \Z)) [t], x \mapsto x t^{|x|/2}</math>

where the left is the cohomology ring of even terms, η is a ring homomorphism that disregards grading and ''x'' is homogeneous and has degree |''x''|.</ref> In particular, <math>c_t(E)</math> is completely determined by the '''total Chern class''' of ''E'': <math>c(E) =1 + c_1(E) + \cdots + c_n(E)</math> and conversely.

The Whitney sum formula, one of the axioms of Chern classes (see below), says that ''c''<sub>''t''</sub> is additive in the sense:
<math display="block">c_t(E \oplus E') = c_t(E) c_t(E').</math>
Now, if <math>E = L_1 \oplus \cdots \oplus L_n</math> is a direct sum of (complex) line bundles, then it follows from the sum formula that:
<math display="block">c_t(E) = (1+a_1(E) t) \cdots (1+a_n(E) t)</math>
where <math>a_i(E) = c_1(L_i)</math> are the first Chern classes. The roots <math>a_i(E)</math>, called the '''Chern roots''' of ''E'', determine the coefficients of the polynomial: i.e.,
<math display="block">c_k(E) = \sigma_k(a_1(E), \ldots, a_n(E))</math>
where σ<sub>''k''</sub> are [[elementary symmetric polynomials]]. In other words, thinking of ''a''<sub>''i''</sub> as formal variables, ''c''<sub>''k''</sub> "are" σ<sub>''k''</sub>. A basic fact on [[symmetric polynomial]]s is that any symmetric polynomial in, say, ''t''<sub>''i''</sub>'s is a polynomial in elementary symmetric polynomials in ''t''<sub>''i''</sub>'s. Either by [[splitting principle]] or by ring theory, any Chern polynomial <math>c_t(E)</math> factorizes into linear factors after enlarging the cohomology ring; ''E'' need not be a direct sum of line bundles in the preceding discussion. The conclusion is
{{block indent | em = 1.5 | text = "One can evaluate any symmetric polynomial ''f'' at a complex vector bundle ''E'' by writing ''f'' as a polynomial in σ<sub>''k''</sub> and then replacing σ<sub>''k''</sub> by ''c''<sub>''k''</sub>(''E'')."}}

'''Example''': We have polynomials ''s''<sub>''k''</sub>
<math display="block">t_1^k + \cdots + t_n^k = s_k(\sigma_1(t_1, \ldots, t_n), \ldots, \sigma_k(t_1, \ldots, t_n))</math>
with <math>s_1 = \sigma_1, s_2 = \sigma_1^2 - 2 \sigma_2</math> and so on (cf. [[Newton's identities#Expressing power sums in terms of elementary symmetric polynomials|Newton's identities]]). The sum
<math display="block">\operatorname{ch}(E) = e^{a_1(E)} + \cdots + e^{a_n(E)} = \sum s_k(c_1(E), \ldots, c_n(E)) / k!</math>
is called the Chern character of ''E'', whose first few terms are: (we drop ''E'' from writing.)
<math display="block">\operatorname{ch}(E) = \operatorname{rk} + c_1 + \frac{1}{2}(c_1^2 - 2c_2) + \frac{1}{6} (c_1^3 - 3c_1c_2 + 3c_3) + \cdots.</math>

'''Example''': The [[Todd class]] of ''E'' is given by:
<math display="block">\operatorname{td}(E) = \prod_1^n {a_i \over 1 - e^{-a_i}} = 1 + {1 \over 2} c_1 + {1 \over 12} (c_1^2 + c_2) + \cdots.</math>

'''Remark''': The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let ''G''<sub>''n''</sub> be the [[infinite Grassmannian]] of ''n''-dimensional complex vector spaces.  This space is equipped with a tautologous vector bundle of rank <math>n</math>, say <math>E_n \to G_n</math>.  <math>G_n</math> is called the [[classifying space]] for rank-<math>n</math> vector bundles because given any complex vector bundle ''E'' of rank ''n'' over ''X'', there is a continuous map 
<math display="block">f_E: X \to G_n</math>
such that the pullback of <math>E_n</math> to <math>X</math> along <math>f_E</math> is isomorphic to <math>E</math>, and this map <math>f_E</math> is  unique up to homotopy. [[Borel's theorem]] says the cohomology ring of ''G''<sub>''n''</sub> is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σ<sub>''k''</sub>; so, the pullback of ''f''<sub>''E''</sub> reads:
<math display="block">f_E^*: \Z [\sigma_1, \ldots, \sigma_n] \to H^*(X, \Z ).</math>
One then puts:
<math display="block">c_k(E) = f_E^*(\sigma_k).</math>

'''Remark''': Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let <math>\operatorname{Vect}_n^{\Complex}</math> 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 <math>\operatorname{Vect}_n^{\Complex } = [-, G_n]</math> to the cohomology functor <math>H^*(-, \Z ).</math> 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''<sub>''n''</sub>:
<math display="block">\operatorname{Nat}([-, G_n], H^*(-, \Z )) = H^*(G_n, \Z ) = \Z [\sigma_1, \ldots, \sigma_n].</math>

== Computation formulae ==
Let ''E'' be a vector bundle of rank ''r'' and <math>c_t(E) = \sum_{i = 0}^r c_i(E)t^i</math> the [[#Chern polynomial|Chern polynomial]] of it.
*For the [[dual bundle]] <math>E^*</math> of <math>E</math>, <math>c_i(E^*) = (-1)^i c_i(E)</math>.<ref>{{harvnb|Fulton|loc=Remark 3.2.3. (a)}}</ref>
*If ''L'' is a line bundle, then<ref>{{harvnb|Fulton|loc=Remark 3.2.3. (b)}}</ref><ref>{{harvnb|Fulton|loc=Example 3.2.2. }}</ref> <math display="block">c_t(E \otimes L) = \sum_{i=0}^r c_i(E) c_t(L)^{r-i} t^i</math> and so <math>c_i(E \otimes L), i = 1, 2, \dots, r</math> are <math display="block">c_1(E) + r c_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.</math>
*For the Chern roots <math>\alpha_1, \dots, \alpha_r</math> of <math>E</math>,<ref>{{harvnb|Fulton|loc=Remark 3.2.3. (c)}}</ref> <math display="block">\begin{align}
c_t(\operatorname{Sym}^p E) &= \prod_{i_1 \le \cdots \le 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{align}</math> In particular, <math>c_1(\wedge^r E) = c_1(E).</math>
*For example,<ref>Use, for example, WolframAlpha to expand the polynomial and then use the fact <math>c_i</math> are elementary symmetric polynomials in <math>\alpha_i</math>'s.</ref> for <math>c_i = c_i(E)</math>,
*:when <math>r = 2</math>, <math>c(\operatorname{Sym}^2 E) = 1 + 3c_1 + 2 c_1^2 + 4 c_2 + 4 c_1 c_2,</math> 
*:when <math>r = 3</math>, <math>c(\operatorname{Sym}^2 E) = 1 + 4c_1 + 5 c_1^2 + 5 c_2 + 2 c_1^3 + 11 c_1 c_2 + 7 c_3.</math>
:(cf. [[Segre class#Example 2]].)

=== Applications of formulae ===
We can use these abstract properties to compute the rest of the chern classes of line bundles on <math>\mathbb{CP}^1</math>. Recall that <math>\mathcal{O}(-1)^* \cong \mathcal{O}(1)</math> showing <math>c_1(\mathcal{O}(1)) = 1 \in H^2(\mathbb{CP}^1;\mathbb{Z})</math>. Then using tensor powers, we can relate them to the chern classes of <math>c_1(\mathcal{O}(n)) = n</math> for any integer.

== Properties ==

Given a [[complex vector bundle]] ''E'' over a [[topological space]] ''X'', the Chern classes of ''E'' are a sequence of elements of the [[cohomology]] of ''X''. The '''''k''-th Chern class''' of ''E'', which is usually denoted ''c<sub>k</sub>''(''E''), is an element of
<math display="block">H^{2k}(X;\Z),</math>
the cohomology of ''X'' with [[integer]] coefficients.  One can also define the '''total Chern class'''
<math display="block">c(E) = c_0(E) + c_1(E) + c_2(E) + \cdots .</math>

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.{{clarify|date=December 2014}}

===Classical axiomatic definition===
The Chern classes satisfy the following four axioms:
# <math>c_0(E) = 1</math> for all ''E''.
# Naturality: If <math>f : Y \to X</math> is [[continuous function (topology)|continuous]] and ''f*E'' is the [[pullback bundle|vector bundle pullback]] of ''E'', then <math>c_k(f^* E) = f^* c_k(E)</math>.
# [[Hassler Whitney|Whitney]] sum formula: If <math>F \to X</math> is another complex vector bundle, then the Chern classes of the [[direct sum of vector bundles|direct sum]] <math>E \oplus F</math> are given by <math display="block">c(E \oplus F) = c(E) \smile c(F);</math> that is, <math display="block">c_k(E \oplus F) = \sum_{i = 0}^k c_i(E) \smile c_{k - i}(F).</math>
# Normalization: The total Chern class of the [[tautological line bundle]] over <math>\mathbb{CP}^k</math> is 1−''H'', where ''H'' is [[Poincaré duality|Poincaré dual]] to the [[hyperplane]] <math>\mathbb{CP}^{k - 1} \subseteq \mathbb{CP}^k</math>.

===Grothendieck axiomatic approach===

Alternatively, {{harvs|txt|authorlink=Alexander Grothendieck|first=Alexander|last=Grothendieck|year=1958}} replaced these with a slightly smaller set of axioms:

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

He shows using the [[Leray–Hirsch theorem]] that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.

Namely, introducing the projectivization <math>\mathbb{P}(E)</math> of the rank ''n'' complex vector bundle ''E'' → ''B'' as the fiber bundle on ''B'' whose fiber at any point <math>b\in B</math> is the projective space of the fiber ''E<sub>b</sub>''. The total space of this bundle <math>\mathbb{P}(E)</math> is equipped with its tautological complex line bundle, that we denote <math>\tau</math>, and the first Chern class
<math display="block">c_1(\tau)=: -a</math>
restricts on each fiber <math>\mathbb{P}(E_b)</math> to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of [[complex projective space]]s.

The classes
<math display="block">1, a, a^2, \ldots , a^{n-1}\in H^*(\mathbb{P}(E))</math>
therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The [[Leray–Hirsch theorem]] then states that any class in <math>H^*(\mathbb{P}(E))</math> can be written uniquely as a linear combination of the 1, ''a'', ''a''<sup>2</sup>, ..., ''a''<sup>''n''−1</sup> with classes on the base as coefficients.

In particular, one may define the Chern classes of ''E'' in the sense of Grothendieck, denoted <math>c_1(E), \ldots c_n(E)</math> by expanding this way the class <math>-a^n</math>, with the relation:
<math display="block"> - a^n = c_1(E)\cdot a^{n-1}+ \cdots + c_{n-1}(E) \cdot a + c_n(E) .</math>

One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.

===The top Chern class===
In fact, these properties uniquely characterize the Chern classes. They imply, among other things:

* If ''n'' is the complex rank of ''V'', then <math>c_k(V) = 0</math> for all ''k'' > ''n''. Thus the total Chern class terminates.
* The top Chern class of ''V'' (meaning <math>c_n(V)</math>, where ''n'' is the rank of ''V'') is always equal to the [[Euler class]] of the underlying real vector bundle.

== In algebraic geometry ==
=== Axiomatic description ===
There is another construction of Chern classes which take values in the algebrogeometric analogue of the cohomology ring, the [[Chow ring]].

Let <math>X</math> be a nonsingular quasi-projective variety of dimension <math>n</math>. It can be shown that there is a unique theory of Chern classes which assigns an algebraic vector bundle <math>E \to X</math> to elements <math>c_i(E) \in A^i(X)</math> called Chern classes, with Chern polynomial <math>c_t(E)=c_0(E) + c_1(E)t + \cdots + c_n(E)t^n</math>, satisfying the following (similar to [[Chern_class#Grothendieck_axiomatic_approach |Grothendieck's axiomatic approach]]). <ref>{{harvnb|Hartshorne|loc=Appendix A. 3 Chern Classes.}}</ref>
# If for a Cartier divisor <math>D</math>, we have <math>E \cong \mathcal{O}_X(D)</math>, then <math>c_t(E) = 1+Dt</math>.
# If <math>f: X' \to X</math> is a morphism, then <math>c_i(f^*E) = f^* c_i(E)</math>.
# If <math>0 \to E' \to E \to E'' \to 0</math> is an exact sequence of vector bundles on <math>X</math>, the Whitney sum formula holds: <math>c_t(E) = c_t(E')c_t(E'')</math>.

<!-- the previously listed axioms were not entirely correct---the Chern classes do NOT give a ring homomorphism from the K-group to the Chow ring (e.g., see the Whitney sum formula turning addition in the K-group into multiplication in the Chow ring, rather than preserving addition). To get a ring homomorphism, you need to use the Chern character instead. -->

<!--
These are nice but are actually already done earlier; merge them with the early section
=== Abstract computations using formal [roperties ===
==== Direct sums of line bundles ====
Using these relations we can make numerous computations for vector bundles. First, notice that if we have line bundles <math>\mathcal{L},\mathcal{L}'</math> we can form a short exact sequence of vector bundles
<math display="block">0 \to \mathcal{L} \to \mathcal{L}\oplus\mathcal{L}' \to \mathcal{L}' \to 0</math>
Using properties <math>1</math> and <math>2</math> we have that
<math display="block">\begin{align}
c(\mathcal{L}\oplus\mathcal{L}') &= c(\mathcal{L})c(\mathcal{L}') \\
&= (1+c_1(\mathcal{L}))(1+c_1(\mathcal{L}')) \\
&= 1 + c_1(\mathcal{L}) + c_1(\mathcal{L}') + c_1(\mathcal{L})c_1(\mathcal{L}') 
\end{align}</math>
By induction, we have
<math display="block">c(\bigoplus^n_{i=1} \mathcal{L}_i) = c(\mathcal{L}_1)\cdots c(\mathcal{L}_n)</math>

====Duals of line bundles====
Since line bundles on a smooth projective variety <math>X</math> are determined by a divisor class <math>[D]</math> and the dual line bundle is determined by the negative divisor class <math>-[D]</math>, we have that
<math display="block">c_1(\mathcal{L}) = -c_1(\mathcal{L}^*)</math>

===Tangent bundle of projective space===
This can be applied to the Euler sequence for projective space
<math display="block">0 \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_{\mathbb{P}^n}(1)^{\oplus (n+1)} \to \mathcal{T}_{\mathbb{P}^n} \to 0</math>
to compute
<math display="block">\begin{align}
c(\mathcal{O}_{\mathbb{P}^n})c(\mathcal{T}_{\mathbb{P}^n}) &= c(\mathcal{O}_{\mathbb{P}^n}(1)^{\oplus (n+1)}) \\
c(\mathcal{T}_{\mathbb{P}^n})&= (1 + H)^{n+1} \\
&= {n+1 \choose 0}1 + {n+1 \choose 1}H + \cdots + {n+1 \choose n}H^n
\end{align}</math>
where <math>H</math> is the class of a degree one hyperplane. Also, notice that <math>H^{n+1}=0</math> in the chow ring of <math>\mathbb{P}^n</math>.
-->

=== Normal sequence ===
Computing the characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety <math>X \subset \mathbb{P}^n</math> there is the short exact sequence
<math display="block">0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^n}|_X \to \mathcal{N}_{X/\mathbb{P}^n} \to 0</math>

==== Quintic threefold ====
For example, consider a nonsingular [[quintic threefold]] in <math>\mathbb{P}^4</math>.  Its normal bundle is given by <math>\mathcal{O}_X(5)</math> and we have the short exact sequence
<math display="block">0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^4}|_X \to \mathcal{O}_X(5) \to 0</math>

Let <math>h</math> denote the hyperplane class in <math>A^\bullet(X)</math>. Then the Whitney sum formula gives us that
<math display="block">c(\mathcal{T}_X)c(\mathcal{O}_X(5)) = (1+h)^5 = 1 + 5h + 10h^2 + 10h^3 </math>

Since the Chow ring of a hypersurface is difficult to compute, we will consider this sequence as a sequence of coherent sheaves in <math>\mathbb{P}^4</math>. This gives us that
<math display="block">\begin{align}
c(\mathcal{T}_X) &= \frac{1 + 5h + 10h^2 + 10h^3}{1 + 5h} \\
&= \left(1 + 5h + 10h^2 + 10h^3\right)\left(1 - 5h + 25h^2 - 125h^3\right) \\
&= 1 + 10h^2 - 40h^3
\end{align}</math>

Using the Gauss-Bonnet theorem we can integrate the class <math>c_3(\mathcal{T}_X)</math> to compute the Euler characteristic. Traditionally this is called the [[Euler class]]. This is
<math display="block">\int_{[X]} c_3(\mathcal{T}_X) = \int_{[X]} -40h^3 = -200</math>
since the class of <math>h^3</math> can be represented by five points (by [[Bézout's theorem]]). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of <math>X</math> by using the definition of the Euler characteristic and using the Lefschetz hyperplane theorem.
<!-- Cite https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf page 181-182 -->

==== Degree d hypersurfaces ====
If <math>X \subset \mathbb{P}^3</math> is a degree <math>d</math> smooth hypersurface, we have the short exact sequence <math display="block">0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^3}|_X \to \mathcal{O}_X(d) \to 0</math> giving the relation <math display="block">c(\mathcal{T}_X) = \frac{c(\mathcal{T}_{\mathbb{P}^3|_X})}{c(\mathcal{O}_X(d))}</math> we can then calculate this as
<math display="block">\begin{align}
c(\mathcal{T}_X) &= \frac{(1+[H])^4}{(1 + d[H])} \\
&= (1 + 4[H] + 6[H]^2)(1-d[H]+d^2[H]^2) \\
&= 1 + (4-d)[H] + (6-4d+d^2)[H]^2
\end{align}</math>
Giving the total chern class. In particular, we can find <math>X</math> is a spin 4-manifold if <math>4-d </math> is even, so every smooth hypersurface of degree <math>2k</math> is a [[spin manifold]].

==Proximate notions==

===The Chern character===
Chern classes can be used to construct a homomorphism of rings from the [[topological K-theory]] of a space to (the completion of) its rational cohomology. For a line bundle ''L'', the Chern character ch is defined by

<math display="block">\operatorname{ch}(L) = \exp(c_1(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.</math>

More generally, if <math>V = L_1 \oplus \cdots \oplus L_n</math> is a direct sum of line bundles, with first Chern classes <math>x_i = c_1(L_i),</math> the Chern character is defined additively
<math display="block"> \operatorname{ch}(V)  = e^{x_1} + \cdots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + \cdots + x_n^m). </math>

This can be rewritten as:<ref>(See also {{slink||Chern polynomial}}.) Observe that when ''V'' is a sum of line bundles, the Chern classes of ''V'' can be expressed as [[elementary symmetric polynomials]] in the <math>x_i</math>, <math>c_i(V) = e_i(x_1,\ldots,x_n).</math>
In particular, on the one hand
<math display="block">c(V) := \sum_{i=0}^n c_i(V),</math>
while on the other hand
<math display="block">\begin{align}
c(V) &= c(L_1 \oplus \cdots \oplus L_n) \\
&= \prod_{i=1}^n c(L_i) \\
&= \prod_{i=1}^n (1+x_i) \\
&= \sum_{i=0}^n e_i(x_1,\ldots,x_n)
\end{align}</math>
Consequently, [[Newton's identities#Expressing power sums in terms of elementary symmetric polynomials|Newton's identities]] may be used to re-express the power sums in ch(''V'') above solely in terms of the Chern classes of ''V'', giving the claimed formula.</ref>

<math display="block"> \operatorname{ch}(V) = \operatorname{rk}(V) + c_1(V) + \frac{1}{2}(c_1(V)^2 - 2c_2(V)) + \frac{1}{6} (c_1(V)^3 - 3c_1(V)c_2(V) + 3c_3(V)) + \cdots.</math>

This last expression, justified by invoking the [[splitting principle]], is taken as the definition ''ch(V)'' for arbitrary vector bundles ''V''.

If a connection is used to define the Chern classes when the base is a manifold (i.e., the [[Chern–Weil theory]]), then the explicit form of the Chern character is
<math display="block">\operatorname{ch}(V)=\left[\operatorname{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)\right]</math>
where {{math|Ω}} is the [[curvature form|curvature]] of the connection.

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product.  Specifically, it obeys the following identities:

<math display="block">\operatorname{ch}(V \oplus W) = \operatorname{ch}(V) + \operatorname{ch}(W)</math>
<math display="block">\operatorname{ch}(V \otimes W) = \operatorname{ch}(V) \operatorname{ch}(W).</math>

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ''ch'' is a [[homomorphism]] of [[abelian group]]s from the [[K-theory]] ''K''(''X'') into the rational cohomology of ''X''.  The second identity establishes the fact that this homomorphism also respects products in ''K''(''X''), and so ''ch'' is a homomorphism of rings.

The Chern character is used in the [[Hirzebruch–Riemann–Roch theorem]].

===Chern numbers===

If we work on an [[orientable manifold|oriented manifold]] of dimension <math>2n</math>, then any product of Chern classes of total degree <math>2n</math> (i.e., the sum of indices of the Chern classes in the product should be <math>n</math>) can be paired with the [[orientation homology class]] (or "integrated over the manifold") to give an integer, a '''Chern number''' of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by <math>c_1^3</math>, <math>c_1 c_2</math>, and <math>c_3</math>.  In general, if the manifold has dimension <math>2n</math>, the number of possible independent Chern numbers is the number of [[integer partition|partition]]s of <math>n</math>.

The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.

===Generalized cohomology theories===

There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a [[generalized cohomology theory]].  The theories for which such generalization is possible are called ''[[Complex cobordism#Formal group laws|complex orientable]]''. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a [[formal group law]].

===Algebraic geometry===

In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in:

*For complex varieties the Chern classes can take values in ordinary cohomology, as above.
*For varieties over general fields, the Chern classes can take values in cohomology theories such as [[etale cohomology]] or [[l-adic cohomology]].
*For varieties ''V'' over general fields the Chern classes can also take values in homomorphisms of [[Chow group]]s CH(V): for example, the first Chern class of a line bundle over a variety ''V'' is a homomorphism from CH(''V'') to CH(''V'') reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the [[cap product]].

=== Manifolds with structure ===

The theory of Chern classes gives rise to [[cobordism]] invariants for [[almost complex manifold]]s.

If ''M'' is an almost complex manifold, then its [[tangent bundle]] is a complex vector bundle.  The '''Chern classes''' of ''M'' are thus defined to be the Chern classes of its tangent bundle.  If ''M'' is also [[Compact space|compact]] and of dimension 2''d'', then each [[monomial]] of total degree 2''d'' in the Chern classes can be paired with the [[fundamental class]] of ''M'', giving an integer, a '''Chern number''' of ''M''. If ''M''′ is another almost complex manifold of the same dimension, then it is cobordant to ''M'' if and only if the Chern numbers of ''M''′ coincide with those of ''M''.

The theory also extends to real [[Symplectic geometry|symplectic]] vector bundles, by the intermediation of compatible almost complex structures. In particular, [[symplectic manifold]]s have a well-defined Chern class.

=== Arithmetic schemes and Diophantine equations ===

(See [[Arakelov geometry]])

== See also ==
* [[Pontryagin class]]
* [[Stiefel–Whitney class]]
* [[Euler class]]
* [[Segre class]]
* [[Schubert calculus]]
* [[Quantum Hall effect]]
* [[Localized Chern class]]

== Notes ==
{{Reflist}}

==References==
* {{Citation | last1=Chern | first1=Shiing-Shen | author-link=Shiing-Shen Chern| title=Characteristic classes of Hermitian Manifolds | year=1946 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=47 | issue=1 | pages=85–121 | doi=10.2307/1969037 |jstor=1969037}}
* {{cite book |last1=Fulton |first1=W. |title=Intersection Theory |date=29 June 2013 |publisher=Springer Science & Business Media |isbn=978-3-662-02421-8 |ref={{harvid|Fulton}} |url=https://books.google.com/books?id=gCXsCAAAQBAJ |language=en}}
* {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=La théorie des classes de Chern | mr=0116023 | year=1958 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=86 | pages=137–154
| doi=10.24033/bsmf.1501 |url= http://www.numdam.org/item?id=BSMF_1958__86__137_0| doi-access=free }}
* {{cite book |last1=Hartshorne |first1=Robin |title=Algebraic Geometry |date=29 June 2013 |publisher=Springer Science & Business Media |isbn=978-1-4757-3849-0  |url=https://books.google.com/books?id=7z4mBQAAQBAJ |ref={{harvid|Hartshorne}} |language=en}}
* {{Citation | last1=Jost | first1=Jürgen | author-link=Jürgen Jost| title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | edition=4th | isbn=978-3-540-25907-7 | year=2005}} (Provides a very short, introductory review of Chern classes).
* {{citation|first=J. Peter |last=May|author-link=J. Peter May| title=A Concise Course in Algebraic Topology|publisher= University of Chicago Press|year= 1999|isbn=9780226511832|url=https://books.google.com/books?id=g8SG03R1bpgC&q=%22Chern+class%22&pg=PA1}}
* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | last2=Stasheff | first2=James D. |author2-link=Jim Stasheff| title=Characteristic classes | publisher=Princeton University Press; University of Tokyo Press | series=Annals of Mathematics Studies | isbn=978-0-691-08122-9 | year=1974 | volume=76}}
* {{Citation | last1=Rubei | first1=Elena |  title=Algebraic Geometry, a concise dictionary | publisher=Walter De Gruyter | isbn=978-3-11-031622-3 | year=2014}}

== External links ==
* [http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory] – A downloadable book-in-progress by [[Allen Hatcher]]. Contains a chapter about characteristic classes.
*[[Dieter Kotschick]], [http://www.physorg.com/news163858041.html Chern numbers of algebraic varieties]

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[[Category:Characteristic classes]]
[[Category:Chinese mathematical discoveries]]