 Vanishing of global sections of the once-twisted projective cotangent bundle

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.