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?




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[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.



Hard Lefschetz theorem in intersection cohomology
For an algebraically closed field, you need to use the étale topology as the analytic topology is not available. You then must use $\mathbb Q\_\ell$-coefficients rather than $\mathbb Q$-coefficients. As far as I know, you must use the formalism of the derived category - I don't know how to define intersection homology in the étale setting without perverse sheaves which are defined via the derived category.


Indeed, the method to deduce the $\mathbb C$ case in section 6 of BBD(G) is to first deduce the $\mathbb C$ case in the étale topology with $\mathbb Q\_\ell$-coefficients and then from that deduce the statement in the analytic topology. The first step works without modification in an arbitrary algebraically closed field of characteristic zero.


