= Vanishing of global sections of the once-twisted projective cotangent bundle
{title2=$H^0(\mathbf P^n,\Omega^1(1))=0$}
Dualize the <Euler sequence on complex projective space> and tensor by the <hyperplane line bundle> to obtain $0\to\Omega^1(1)\to\mathcal O^{n+1}\to\mathcal O(1)\to0$. On global sections the last map sends $(c_i)$ to $\sum_i c_iX_i$, 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.
Back to article page
