diagram\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now,\n\nH 2 ( X, Z ) / / H 2 ( X, O X ) ‚âÉ Dolbeault H 2 ( X, C ) deRham ‚âÉ H 2 dR ( X, C ) / / H 0 , 2 ¬Ø ‚àÇ ( X )\n\nof the proof follows as the ( 1 , 1 ) -Lefschetz theorem in [6].\n\nRemark 3.5 . For k = 1 and P d Œ£ 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 ) ‚âÉ H dim X ‚àí 1 , dim X ‚àí 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 Hard Lefschetz theorem for projective orbifolds. The dimension 2 and 3 cases are covered by Theorem 3.5 and the Hard Lefschetz.\n\nCayley trick and Cayley proposition\n\nThe Cayley trick is a way to associate to a quasi-smooth intersection subvariety a quasi- smooth hypersurface. Let L 1 , . . . , L s be line bundles on P d Œ£ and let œÄ ‚à∂ P ( E ) ‚Üí P d Œ£ be the projective space bundle associated to the vector bundle E = L 1 ‚äï ‚ãØ ‚äï L s . It is known that P ( E ) is a ( d + s ‚àí 1 ) -dimensional simplicial toric variety whose fan depends on the degrees of the line bundles and the fan Œ£. Furthermore, if the Cox ring, without considering the grading, of P d Œ£ is C [ x 1 , . . . , x m ] then the Cox ring of P ( E ) is\n\nMoreover for X a quasi-smooth intersection subvariety cut oÔ¨Ä by f 1 , . . . , f s with deg ( f i ) = [ L i ] we relate the hypersurface Y cut oÔ¨Ä by F = y 1 f 1 + ‚ãÖ ‚ãÖ ‚ãÖ + y s f s which turns out to be quasi-smooth. For more details see Section 2 in [7].\n\nWe will denote P ( E ) as P d + s ‚àí 1 Œ£ ,X to keep track of its relation with X and P d Œ£ .\n\nThe following is a
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. ->: diagram\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now,\n\nH 2 ( X, Z ) / / H 2 ( X, O X ) ‚âÉ Dolbeault H 2 ( X, C ) deRham ‚âÉ H 2 dR ( X, C ) / / H 0 , 2 ¬Ø ‚àÇ ( X )\n\nof the proof follows as the ( 1 , 1 ) -Lefschetz theorem in [6].\n\nRemark 3.5 . For k = 1 and P d Œ£ 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 ) ‚âÉ H dim X ‚àí 1 , dim X ‚àí 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 Hard Lefschetz theorem for projective orbifolds. The dimension 2 and 3 cases are covered by Theorem 3.5 and the Hard Lefschetz.\n\nCayley trick and Cayley proposition\n\nThe Cayley trick is a way to associate to a quasi-smooth intersection subvariety a quasi- smooth hypersurface. Let L 1 , . . . , L s be line bundles on P d Œ£ and let œÄ ‚à∂ P ( E ) ‚Üí P d Œ£ be the projective space bundle associated to the vector bundle E = L 1 ‚äï ‚ãØ ‚äï L s . It is known that P ( E ) is a ( d + s ‚àí 1 ) -dimensional simplicial toric variety whose fan depends on the degrees of the line bundles and the fan Œ£. Furthermore, if the Cox ring, without considering the grading, of P d Œ£ is C [ x 1 , . . . , x m ] then the Cox ring of P ( E ) is\n\nMoreover for X a quasi-smooth intersection subvariety cut oÔ¨Ä by f 1 , . . . , f s with deg ( f i ) = [ L i ] we relate the hypersurface Y cut oÔ¨Ä by F = y 1 f 1 + ‚ãÖ ‚ãÖ ‚ãÖ + y s f s which turns out to be quasi-smooth. For more details see Section 2 in [7].\n\nWe will denote P ( E ) as P d + s ‚àí 1 Œ£ ,X to keep track of its relation with X and P d Œ£ .\n\nThe following is a