Chern class axioms in the Chow ring ($c_i(E)\in {\mathrm{CH}}^i(X)$)

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.