We first generalize the Euler exact sequence, allowing to obtain an explicit and tractable description of cohomology groups of twisted symmetric powers of cotangent bundles of projective spaces. Then, via a particular rank \(2\) vector bundle on the projectivized tangent bundle \(\textbf{P}(T\textbf{P}^{N})\) , we obtain a resolution of the structural sheaf of the projectivized tangent bundle of a smooth complete intersection \(Z \subset \textbf{P}^{N}\) . Combining these two facts, we obtain explicit descriptions of cohomology groups of twisted symmetric powers of cotangent bundles of smooth complete intersections, which, in some particular cases, are simple enough to be fully understood. As a first application, we prove a non-vanishing theorem, which shows that the classic vanishing theorem of Bruckmann–Rackvitz is optimal. As a second application, we study explicitly the algebra \(\begin{aligned} \bigoplus _{m \in \mathbb {N}} H^{0}(Z, S^{m}\Omega _{Z}(m)), \end{aligned}\) where \(Z\) is a smooth complete intersection whose dimension is greater than its codimension.
