# Chern class

Chern classes are characteristic classes assigned to complex vector bundles: for a complex vector bundle E of rank r over a space M, the i-th Chern class c_i(E) is a cohomology class in H^{2i}(M; Z), and together they measure how the bundle is twisted globally. They were introduced by [Shiing-Shen Chern](https://www.edgepedia.org/shiing-shen-chern) in his 1946 [Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics) paper "Characteristic classes of Hermitian manifolds" (Annals of Mathematics 47(2), pp. 85–121), built from the curvature forms of a connection.<sup>[1](https://celebratio.org/Chern_SS/article/286/)</sup><sup> • </sup><sup>[2](https://explore.metascienceobservatory.org/papers/W2331812412)</sup>

| Key fact | Detail |
|---|---|
| Definition | c_i(E) ∈ H^{2i}(M; Z) for i = 0, ..., r; the total Chern class is c(E) = c_0(E) + c_1(E) + ... + c_r(E) with c_0 = 1 and c_i = 0 for i > rank(E)<sup>[3](https://www.math.purdue.edu/~adebray/lecture_notes/u17_characteristic_classes.pdf)</sup><sup> • </sup><sup>[4](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)</sup> |
| Classifying space | H*(BU(n); Z) ≅ Z[c_1, ..., c_n], a polynomial algebra on n generators; every characteristic class of complex n-plane bundles is a polynomial in the Chern classes<sup>[5](https://math.stanford.edu/~ralph/fiber.pdf)</sup> |
| Whitney sum formula | c(E ⊕ F) = c(E)·c(F), extended to short exact sequences 0 → F → E → G → 0<sup>[6](https://www.math.columbia.edu/~chaoli/docs/IntersectionTheory.html)</sup> |
| Splitting principle | E can be pulled back so that it splits into line bundles with c(E) = ∏(1 + c_1(L_i)); any Chern class formula valid for sums of line bundles holds in general<sup>[6](https://www.math.columbia.edu/~chaoli/docs/IntersectionTheory.html)</sup> |
| Chern-Weil formula | c_k(E) = [P_k(iF_∇/2π)] where F_∇ is the curvature of a connection; the class is independent of the connection<sup>[4](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)</sup> |
| Projective space | c(T_CP^n) = (1 + H)^{n+1} with H = c_1(O(1)) the standard generator of H^2(CP^n, Z)<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup> |
| Algebraic version | In the Stacks project, Chern classes of a rank r bundle are defined in the Chow groups via the projective space bundle formula; for a line bundle, c_1 equals the associated Weil divisor<sup>[8](https://stacks.math.columbia.edu/tag/02TZ)</sup> |

## Definition and axioms

For each complex vector bundle E → M of rank r, the Chern classes are natural cohomology classes c_i(E) ∈ H^{2i}(M; Z), written c_i(M) for the tangent bundle and c(M) for its total class. The axiomatic definition is due to [Alexander Grothendieck](https://www.edgepedia.org/alexander-grothendieck).<sup>[3](https://www.math.purdue.edu/~adebray/lecture_notes/u17_characteristic_classes.pdf)</sup> Naturality means a map f: N → M pulls back c_i(E) to c_i(f*E).<sup>[3](https://www.math.purdue.edu/~adebray/lecture_notes/u17_characteristic_classes.pdf)</sup>

The classes are complete in a precise sense: the ring of unitary characteristic classes is the polynomial algebra Z[c_1, ..., c_n] on the universal classes over the classifying space BU(n), so every characteristic class of complex n-plane bundles is a unique polynomial in the Chern classes.<sup>[5](https://math.stanford.edu/~ralph/fiber.pdf)</sup> The total Chern class packages them as c(η ⊕ ξ) = c(η)c(ξ), the Whitney product formula.<sup>[9](https://www.math.uchicago.edu/~may/CHAR/charclasses.pdf)</sup>

**First Chern class.** The class c_1 has a bundle-theoretic interpretation: it is the obstruction to reducing the structure group of a complex n-plane bundle from U(n) to SU(n).<sup>[9](https://www.math.uchicago.edu/~may/CHAR/charclasses.pdf)</sup> For line bundles it is the unique characteristic class of principal U(1)-bundles, and complex line bundles over a space are classified by their first Chern class.<sup>[10](https://ncatlab.org/nlab/show/Chern%2Bclass)</sup><sup> • </sup><sup>[11](https://www.ocw.mit.edu/courses/18-966-geometry-of-manifolds-spring-2007/resources/lect11/)</sup> For a rank r bundle, c_1(E) equals the first Chern class of its top exterior power Λ^r E.<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup>

## Splitting principle and Chern roots

The splitting principle states that for any vector bundle E → X there is a morphism π: Y → X such that π* is injective on cohomology and π*E admits a filtration by subbundles with line bundle quotients L_i, so that c(E) = ∏(1 + c_1(L_i)).<sup>[6](https://www.math.columbia.edu/~chaoli/docs/IntersectionTheory.html)</sup> This lets one compute as if E were a direct sum of line bundles: any formula for Chern classes that holds for such sums holds in general.<sup>[6](https://www.math.columbia.edu/~chaoli/docs/IntersectionTheory.html)</sup>

The formal c_1(L_i) are the Chern roots x_i of E, with c(E) = ∏(1 + x_i). For the tangent bundle of a complex manifold X of complex dimension n one writes c(TX) = ∏_{i=1}^{n}(1 + x_i).<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup> As an example of the method, for a rank 2 bundle E = L ⊕ M with c_1(L) = α and c_1(M) = β, one gets c_1(E) = α + β and c_2(E) = αβ.<sup>[6](https://www.math.columbia.edu/~chaoli/docs/IntersectionTheory.html)</sup>

## Chern-Weil theory

Chern's original construction used curvature. Given a connection ∇ on a complex vector bundle E → M of rank r, the curvature F_∇ is a section of A^2(End(E)). The invariant polynomials P_k in the matrix B defined by det(id + B) = 1 + P_1(B) + ... + P_r(B) produce closed forms P_k(F_∇) whose de Rham cohomology classes are independent of the choice of connection, and the k-th Chern class is defined as c_k(E) = [P_k(iF_∇/2π)].<sup>[4](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)</sup> The same Chern-Weil homomorphism produces the Chern character (from B ↦ tr(e^B)) and the Todd class (from B ↦ det(tB)/det(id − e^{−tB})); both can be expressed as polynomials in the Chern classes, and neither necessarily vanishes above the rank of E.<sup>[4](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)</sup>

For line bundles, several descriptions of c_1 agree in H^2(X, C): the connecting homomorphism from the exponential sequence, the Chern-Weil form [iF_∇/2π], the Atiyah class in H^1(X, Ω_X), and the Poincaré dual of the divisor D when L = O_X(D).<sup>[4](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)</sup>

## The algebraic definition

In algebraic geometry, Chern classes live in the Chow groups rather than cohomology. The Stacks project defines them for a finite locally free sheaf E of rank r on an integral scheme X using the projective space bundle π: P → X associated to E: the classes c_i ∈ CH_{n−i}(X), with c_0 = [X], are characterized by the relation ∑_{i=0}^{r} (−1)^i c_1(O_P(1))^i ∩ π*c_{r−i} = 0, and c_i(E) ∩ [X] = c_i.<sup>[8](https://stacks.math.columbia.edu/tag/02TZ)</sup> For a line bundle this recovers the earlier notion: the first Chern class equals the Weil divisor associated to the invertible sheaf.<sup>[8](https://stacks.math.columbia.edu/tag/02TZ)</sup> The same framework extends to the splitting principle, tensor product formulas such as c_1(E ⊗ F) = r·c_1(F) + s·c_1(E) for ranks r and s, and Chern classes of perfect complexes.<sup>[12](https://stacks.math.columbia.edu/download/chow.pdf)</sup>

## Example: projective space

The tangent bundle of CP^n fits in the Euler sequence 0 → O → O(1)^{⊕(n+1)} → T_CP^n → 0, which follows from the presentation CP^n = (C^{n+1} \ {0})/C*.<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup> The Whitney formula then gives c(T_CP^n) = c(O(1))^{n+1}/c(O) = (1 + H)^{n+1}, where H = c_1(O(1)) is the standard generator of H^2(CP^n, Z).<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup> The source cautions that H repeated n+1 times are not the Chern roots of T_CP^n, since that would give the wrong rank.<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup>

Integrating the top class gives a topological quantity: since H^n integrates to 1, the top Chern class is c_n = (n+1 choose n)H^n, and ∫ c_n = n + 1, which equals the [Euler characteristic](https://www.edgepedia.org/euler-characteristic) χ(CP^n) = n + 1; the top Chern class of a bundle of rank equal to the manifold dimension counts the zeroes of a holomorphic section.<sup>[13](https://people.maths.ox.ac.uk/cmi/library/monographs/cmim01c.pdf)</sup> For a smooth hypersurface X of degree d in P^n, the adjunction sequence gives c(X) = (1 + x)^{n+1}/(1 + dx).<sup>[13](https://people.maths.ox.ac.uk/cmi/library/monographs/cmim01c.pdf)</sup>

## Derived classes and applications

The Chern character repackages Chern class data: for a rank r bundle W with Chern roots y_i, ch(W) = e^{y_1} + ... + e^{y_r} in H^{even}(X, Q), the rational coefficients being needed for the denominators in the exponential.<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup> It is additive on short exact sequences, multiplicative on tensor products (ch(W_1 ⊗ W_2) = ch(W_1)ch(W_2)), and commutes with pullback; on CP^n the Euler sequence gives ch(T_CP^n) = (n+1)e^H − 1.<sup>[7](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)</sup>

These classes feed into index formulas. The Hirzebruch-Riemann-Roch theorem states that for a holomorphic vector bundle E of rank r on a compact complex n-fold, the Euler characteristic χ(E) = ∑(−1)^i h^i(X, E) equals ∫_X ch(E) · td(X), the product of the Chern character and the Todd class of the tangent bundle; for a line bundle on a [Riemann surface](https://www.edgepedia.org/riemann-surface) this recovers h^0(L) − h^1(L) = deg(L) + 1 − g.<sup>[4](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)</sup> The Whitney duality theorem, c(τ)c(ν) = 1 for the tangent and normal bundles of a complex manifold embedded in C^q, follows from the sum being trivial, and nonzero normal Chern classes obstruct embeddings.<sup>[9](https://www.math.uchicago.edu/~may/CHAR/charclasses.pdf)</sup>

## History

Chern classes arose in the 1930s and 1940s from work of [Hassler Whitney](https://www.edgepedia.org/hassler-whitney), Eduard Stiefel, and Shiing-Shen Chern on natural cohomology classes of tangent bundles, sphere bundles, and complex vector bundles.<sup>[14](https://math.uchicago.edu/~epesikoff/charclasses.pdf)</sup> Chern's route ran through an intrinsic proof of the generalized Gauss-Bonnet theorem, out of which came his construction of the characteristic classes of complex vector bundles from curvature forms of a connection.<sup>[15](https://celebratio.org/Chern_SS/article/454/)</sup> His 1946 Annals paper remains the founding reference, and the axiomatic treatment later given by Grothendieck and the classifying-space description over BU(n) are the two standard modern foundations.<sup>[2](https://explore.metascienceobservatory.org/papers/W2331812412)</sup><sup> • </sup><sup>[3](https://www.math.purdue.edu/~adebray/lecture_notes/u17_characteristic_classes.pdf)</sup><sup> • </sup><sup>[5](https://math.stanford.edu/~ralph/fiber.pdf)</sup>

## References

1. [celebratio mathematica — chern — wu on chern](https://celebratio.org/Chern_SS/article/286/)
2. [Characteristic Classes of Hermitian Manifolds, Shiing-Shen Chern, Annals of Mathematics, 1946, Metascience Observatory record](https://explore.metascienceobservatory.org/papers/W2331812412)
3. [Debray, Arun. Characteristic Classes lecture notes, Purdue University](https://www.math.purdue.edu/~adebray/lecture_notes/u17_characteristic_classes.pdf)
4. [Chern-Weil Theory notes, University of Texas at Austin](https://web.ma.utexas.edu/users/gdavtor/notes/cxg.pdf)
5. [Cohen, Ralph. "Chern Classes and Stiefel-Whitney Classes," Stanford University](https://math.stanford.edu/~ralph/fiber.pdf)
6. [Li, Chao. "Intersection theory in algebraic geometry," Columbia University](https://www.math.columbia.edu/~chaoli/docs/IntersectionTheory.html)
7. [Borisov, Lev. "Adventures in Algebraic Geometry," Rutgers University](https://sites.math.rutgers.edu/~borisov/AAG/Adventures%20in%20AG.pdf)
8. [Section 42.37: The Chern classes of a vector bundle, The Stacks Project](https://stacks.math.columbia.edu/tag/02TZ)
9. [May, J. Peter. "Chern Classes," University of Chicago](https://www.math.uchicago.edu/~may/CHAR/charclasses.pdf)
10. [Chern class, nLab](https://ncatlab.org/nlab/show/Chern%2Bclass)
11. [Lecture 11, Geometry of Manifolds, MIT OpenCourseWare](https://www.ocw.mit.edu/courses/18-966-geometry-of-manifolds-spring-2007/resources/lect11/)
12. [Chow Homology and Chern Classes, The Stacks Project (full chapter PDF)](https://stacks.math.columbia.edu/download/chow.pdf)
13. [The Chern Class of P^n and the Euler Sequence, Clay Mathematics Monographs](https://people.maths.ox.ac.uk/cmi/library/monographs/cmim01c.pdf)
14. [Peskikoff et al., "Characteristic Classes: An Introduction," University of Chicago](https://math.uchicago.edu/~epesikoff/charclasses.pdf)
15. [Wu, Wen-tsün. "Chern" (Life & Math commentary), Celebratio Mathematica](https://celebratio.org/Chern_SS/article/454/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Divisors, cycles, and motives › Intersection theory*

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