  In this paper, we provide two different resolutions of structural sheaves of
projectivized tangent bundles of smooth complete intersections. These
resolutions allow in particular to obtain convenient (and completely explicit)
descriptions of cohomology of twisted symmetric powers of cotangent bundles of
complete intersections, which are easily implemented on computer. We then
provide several applications. First, we recover the known vanishing theorems on
the subject, and show that they are optimal via some non-vanishing theorems.
Then, we study the symmetric algebra of global sections of symmetric powers of
$\Omega_X(1)$, where $X$ is a smooth complete intersection of codimension $c <
N/2$, improving the known results in the literature. We also study partial
ampleness of cotangent bundles of general hypersurfaces. Finally, we illustrate
how the explicit descriptions of cohomology can be implemented on computer. In
particular, this allows to exhibit new and simple examples of family of
surfaces along which the canonically twisted pluri-genera do not remain
constant.
