Chern class ($c_i(E)$)

Chern classes are characteristic classes of vector bundles satisfying functoriality and the Whitney product formula $c(E)=c(E^{\prime })c(E^{\prime \prime })$ for a short exact sequence.

For a vector bundle $E$ of rank $r$ on a smooth algebraic variety, algebraic Chern classes have $c_0(E)=1$, vanish for $i>r$, commute with pullback, and obey the Whitney sum formula for Chern classes for every short exact sequence of vector bundles. A line bundle $\mathcal{O}(D)$ has total class $1+[D]$. The projective-bundle splitting construction makes pullback injective and supplies a filtration by line bundles, so these axioms determine all classes as elementary symmetric functions of their first classes. In particular, $c_i(E^{\ast })=(-1{)}^i{c}_i(E)$ and $c_1(L\otimes M)=c_1(L)+c_1(M)$ in the ordinary Chow ring.

For a smooth degree-$d$ projective hypersurface $X\subset {\mathbb{P}}^n$, put $h=c_1({\mathcal{O}}_X(1))$. The cotangent Euler sequence in homogeneous coordinates gives $c({\Omega }_{{\mathbb{P}}^n}^1{|}_X)=(1-h{)}^{n+1}$. The Conormal exact sequence for Kähler differentials and Whitney sum formula for Chern classes therefore yield $c({\Omega }_X^1)=(1-h{)}^{n+1}/(1-dh)$. In codimension $i\le n-1$, its coefficient is $c_i({\Omega }_X^1)={\sum }_{j=0}^i(-1{)}^j(\genfrac{}{}{0ex}{}{n+1}{j})d^{i-j}{h}^i$. Higher classes vanish by rank and dimension; the formal quotient is truncated in the Chow ring.

Here the $x_j$ are the formal Chern roots of a vector bundle. The degree-four term is $(c_1^2+c_2)/12$. Applied to the tangent bundle of a compact complex surface, Riemann–Roch for the structure sheaf gives the Noether formula.

The complex orientation of complex cobordism defines Chern classes of actual complex bundles by the splitting principle and elementary symmetric expressions in the first Chern classes of line summands. For an actual rank-$n$ complex bundle, its top Chern class is its Euler class in a generalized cohomology theory. Chern classes extend stably by the Whitney formula; this stable extension must not be confused with the Euler class of a merely stably complex real bundle.

The relation uses $x=c_1(S)$ for the tautological line on the projectivization of $E$. It follows from the rank of the complementary bundle and Whitney multiplication. The alternating signs are ring subtraction; they do not assert that the First Chern class of a dual line is the additive negative in complex cobordism, where duality uses the formal-group inverse.

// Target: geometry-and-topology.bigb

On a compact oriented $2r$-manifold without boundary, an integral product of Chern classes of total complex degree $r$ pairs with the fundamental class to give a Chern number. A unitary connection supplies Chern-Weil theory representatives, but the integer is independent of that connection. The Second Chern number on a four-manifold is one important example; the integral of the First Chern class over a closed surface is another.

The Total Chern class of a direct sum of complex vector bundles is the product of their total Chern classes. In each degree, $c_k(E\oplus F)={\sum }_{i+j=k}{c}_i(E)⌣c_j(F)$. The degree-zero class is one, classes beyond the bundle rank vanish, and a trivial bundle has total class one.

For a rank-$r$ complex vector bundle, let $h=c_1(S^{\ast })$ for the tautological line $S$ on its projective bundle. The Leray-Hirsch theorem gives the free basis $1,h,\dots ,h^{r-1}$ over base cohomology. The uniquely determined coefficients in the displayed monic relation define the integral Chern classes. The line normalization uses the Euler class of the canonically oriented underlying real two-plane bundle, and uniqueness makes this definition intrinsic and natural under pullback. No choice of a trivializing cover or classifying map remains in the definition.

The total Chern class is $c(E)=1+c_1(E)+c_2(E)+\cdots$.

On projective space, the Euler sequence is

For a degree-zero rational function regular on an open subset of projective space, each derivative $\partial f/\partial X_i$ is a regular homogeneous rational function of degree $-1$. These derivatives define a derivation with values in $\mathcal{O}(-1{)}^{\oplus (n+1)}$, hence a map from the Kähler differential sheaf. The last map is $(g_i)\mapsto {\sum }_i{X}_i{g}_i$. On $X_i\ne 0$, put $y_j=X_j/X_i$ and use frame $X_i^{-1}$. The derivative of $y_j$ has coordinate vector with $1$ at $j$, $-y_j$ at $i$ and zero elsewhere. Those vectors form a free basis of the kernel of $(b_i)\mapsto b_i+{\sum }_{j\ne i}{y}_j{b}_j$, proving exactness directly in every characteristic.

Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain $0\to {\Omega }^1(1)\to {\mathcal{O}}^{n+1}\to \mathcal{O}(1)\to 0$. On global sections the last map sends $(c_i)$ to ${\sum }_i{c}_i{X}_i$, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.