  We prove that the tangent and the reflexivized cotangent sheaves of any
normal projective klt Calabi-Yau or irreducible holomorphic symplectic variety
are not pseudoeffective, generalizing results of A. H\"oring and T. Peternell
arXiv:1710.06183v2 [math.AG]. We provide examples of Calabi-Yau varieties of
small dimension with singularities in codimension 2.
