with X and P d Œ£ .\n\nThe following is a key remark.\n\nRemark 4.1 . There is a morphism Œπ ‚à∂ X ‚Üí Y ‚äÇ P d + s ‚àí 1 Œ£ ,X . Moreover every point z ‚à∂ = ( x, y ) ‚àà Y with y ‚â† 0 has a preimage. Hence for any subvariety W = V ( I W ) ‚äÇ X ‚äÇ P d Œ£ there exists W ‚Ä≤ ‚äÇ Y ‚äÇ P d + s ‚àí 1 Œ£ ,X such that œÄ ( W ‚Ä≤ ) = W , i.e., W ‚Ä≤ = { z = ( x, y ) ‚à£ x ‚àà W } .\n\nFor X ‚äÇ P d Œ£ a quasi-smooth intersection variety the morphism in cohomology induced by the inclusion i ‚àó ‚à∂ H d ‚àí s ( P d Œ£ , C ) ‚Üí H d ‚àí s ( X, C ) is injective by Proposition 1.4 in [7].\n\nDeÔ¨Ånition 4.2. The primitive cohomology of H d ‚àí s prim ( X ) is the quotient H d ‚àí s ( X, C )/ i ‚àó ( H d ‚àí s ( P d Œ£ , C )) and H d ‚àí s prim ( X, Q ) with rational coeÔ¨Écients.\n\nH d ‚àí s ( P d Œ£ , C ) and H d ‚àí s ( X, C ) have pure Hodge structures, and the morphism i ‚àó is com- patible with them, so that H d ‚àí s prim ( X ) gets a pure Hodge structure.\n\nThe next Proposition is the Cayley proposition.\n\nProposition 4.3. [Proposition 2.3 in [3] ] Let X = X 1 ‚à©‚ãÖ ‚ãÖ ‚ãÖ‚à© X s be a quasi-smooth intersec- tion subvariety in P d Œ£ cut oÔ¨Ä by homogeneous polynomials f 1 . . . f s . Then for p ‚â† d + s ‚àí 1 2 , d + s ‚àí 3 2\n\nRemark 4.5 . The above isomorphisms are also true with rational coeÔ¨Écients since H ‚óè ( X, C ) = H ‚óè ( X, Q ) ‚äó 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 + ‚ãØ + y k f k = 0 } ‚äÇ P 2 k + 1 Œ£ ,X be the quasi-smooth hypersurface associated to the quasi-smooth intersection surface X = X f 1 ‚à© ‚ãÖ ‚ãÖ ‚ãÖ ‚à© X f k ‚äÇ P k + 2 Œ£ . 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 ) ‚â† 0. By the Cayley proposition H k,k prim ( Y, Q ) ‚âÉ H 1 , 1 prim ( X, Q ) and by the ( 1 , 1 ) -Lefschetz theorem for projective\n\ntoric orbifolds there is a non-zero algebraic
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ->: with X and P d Œ£ .\n\nThe following is a key remark.\n\nRemark 4.1 . There is a morphism Œπ ‚à∂ X ‚Üí Y ‚äÇ P d + s ‚àí 1 Œ£ ,X . Moreover every point z ‚à∂ = ( x, y ) ‚àà Y with y ‚â† 0 has a preimage. Hence for any subvariety W = V ( I W ) ‚äÇ X ‚äÇ P d Œ£ there exists W ‚Ä≤ ‚äÇ Y ‚äÇ P d + s ‚àí 1 Œ£ ,X such that œÄ ( W ‚Ä≤ ) = W , i.e., W ‚Ä≤ = { z = ( x, y ) ‚à£ x ‚àà W } .\n\nFor X ‚äÇ P d Œ£ a quasi-smooth intersection variety the morphism in cohomology induced by the inclusion i ‚àó ‚à∂ H d ‚àí s ( P d Œ£ , C ) ‚Üí H d ‚àí s ( X, C ) is injective by Proposition 1.4 in [7].\n\nDeÔ¨Ånition 4.2. The primitive cohomology of H d ‚àí s prim ( X ) is the quotient H d ‚àí s ( X, C )/ i ‚àó ( H d ‚àí s ( P d Œ£ , C )) and H d ‚àí s prim ( X, Q ) with rational coeÔ¨Écients.\n\nH d ‚àí s ( P d Œ£ , C ) and H d ‚àí s ( X, C ) have pure Hodge structures, and the morphism i ‚àó is com- patible with them, so that H d ‚àí s prim ( X ) gets a pure Hodge structure.\n\nThe next Proposition is the Cayley proposition.\n\nProposition 4.3. [Proposition 2.3 in [3] ] Let X = X 1 ‚à©‚ãÖ ‚ãÖ ‚ãÖ‚à© X s be a quasi-smooth intersec- tion subvariety in P d Œ£ cut oÔ¨Ä by homogeneous polynomials f 1 . . . f s . Then for p ‚â† d + s ‚àí 1 2 , d + s ‚àí 3 2\n\nRemark 4.5 . The above isomorphisms are also true with rational coeÔ¨Écients since H ‚óè ( X, C ) = H ‚óè ( X, Q ) ‚äó 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 + ‚ãØ + y k f k = 0 } ‚äÇ P 2 k + 1 Œ£ ,X be the quasi-smooth hypersurface associated to the quasi-smooth intersection surface X = X f 1 ‚à© ‚ãÖ ‚ãÖ ‚ãÖ ‚à© X f k ‚äÇ P k + 2 Œ£ . 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 ) ‚â† 0. By the Cayley proposition H k,k prim ( Y, Q ) ‚âÉ H 1 , 1 prim ( X, Q ) and by the ( 1 , 1 ) -Lefschetz theorem for projective\n\ntoric orbifolds there is a non-zero algebraic