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Firstly we show a generalization of the (1,1)-Lefschetz theorem forprojective toric orbifolds and secondly we prove that on 2k-dimensionalquasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces,under the Cayley trick, every rational (k,k)-cohomology class is algebraic,i.e., the Hodge conjecture holds on them.

A Weak (k,k)-Lefschetz Theorem for Projective Toric Orbifolds