In [1,2] the authors proved the Hard Lefschetz theorem in intersection cohomology:
Let $Z$ be a complex projective variety of pure complex dimension $d$, with $\xi\in H^2(Z,\mathbb{Q})$ the first Chern class of an ample line bundle over $Z$. Then there are isomorphisms given by cup product $$ \xi^k\cdot\_\colon IH^{d-k}(Z,\mathbb{Q})\to IH^{d+k}(Z,\mathbb{Q}) $$ for any integer $k>0$.
Question: Does a Hard Lefschetz theorem in intersection cohomology hold for projective varieties of pure dimension over algebraically closed field of characteristic $0$? If it is possible: does one need the formalism of derived categories or something like this?
[1] A. A. Beilinson, J. N. Bernstein, P. Deligne (O. Gabber) - Faisceaux pervers, Astérisque 100 (1982), Soc. Math. Fr.
[2] M. A. A. De Cataldo, L. Migliorini - The Hodge theory of algebraic maps, Ann. Sci. École Norm. Sup., Serie 4, 38 (2005) 693-750.