= Chern classes of a smooth hypersurface cotangent bundle
{c}
{title2=$c(\Omega_X^1)=(1-h)^{n+1}/(1-dh)$}
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 $c(\Omega^1_{\mathbb P^n}|_X)=(1-h)^{n+1}$. 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 $c_i(\Omega_X^1)=\sum_{j=0}^i(-1)^j\binom{n+1}{j}d^{i-j}h^i$. Higher classes vanish by rank and dimension; the formal quotient is truncated in the <Chow ring>.
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