⋅ ⋅∩ X s be a quasi-smooth intersec- tion subvariety in P d Σ cut oﬀ by homogeneous polynomials f 1 . . . f s . Then for p ≠ d + s − 1 2 , d + s − 3 2\n\nRemark 4.5 . The above isomorphisms are also true with rational coeﬃcients since H ● ( X, C ) = H ● ( X, Q ) ⊗ Q C . See the beginning of Section 7.1 in [10] for more details.\n\nTheorem 5.1. Let Y = { F = y 1 f 1 + ⋯ + y k f k = 0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to the quasi-smooth intersection surface X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f k ⊂ P k + 2 Σ . Then on Y the Hodge conjecture holds.\n\nthe Hodge conjecture holds.\n\nProof. If H k,k prim ( X, Q ) = 0 we are done. So let us assume H k,k prim ( X, Q ) ≠ 0. By the Cayley proposition H k,k prim ( Y, Q ) ≃ H 1 , 1 prim ( X, Q ) and by the ( 1 , 1 ) -Lefschetz theorem for projective\n\ntoric orbifolds there is a non-zero algebraic basis λ C 1 , . . . , λ C n with rational coeﬃcients of H 1 , 1 prim ( X, Q ) , that is, there are n ∶ = h 1 , 1 prim ( X, Q )
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