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In this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. As a first application, we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Then, as our main application, we construct varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.

Symmetric differential forms on complete intersection varieties and applications