1 (O ∗ X ) → H 2 ( X, Z ) → H 2 (O X ) ≃ H 0 , 2 ( X )\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now, it is enough to prove the commutativity of the next diagram\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now,\n\nH 2 ( X, Z ) / / H 2 ( X, O X ) ≃ Dolbeault H 2 ( X, C ) deRham ≃ H 2 dR ( X, C ) / / H 0 , 2 ¯ ∂ ( X )\n\nof the proof follows as the ( 1 , 1 ) -Lefschetz theorem in [6].\n\nRemark 3.5 . For k = 1 and P d Σ as the projective space, we recover the classical ( 1 , 1 ) - Lefschetz theorem.\n\nBy the Hard Lefschetz Theorem for projective orbifolds (see [11] for details) we\n\nBy the Hard Lefschetz Theorem for projective orbifolds (see [11] for details) we get an isomorphism of cohomologies :\n\ngiven by the Lefschetz morphism and since it is a morphism of Hodge structures, we have:\n\nH 1 , 1 ( X, Q ) ≃ H dim X − 1 , dim X − 1 ( X, Q )\n\nCorollary 3.6. If the dimension of X is 1 , 2 or 3 . The Hodge conjecture holds on X\n\nProof. If the dim C X = 1 the result is clear by the
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