the group Hom ( Cl ( Œ£ ) , C ‚àó ) and the group action is induced by the Cl ( Œ£ ) - grading of S .\n\nNow we give a brief introduction to complex orbifolds and we mention the needed theorems for the next section. Namely: de Rham theorem and Dolbeault theorem for complex orbifolds.\n\nDeÔ¨Ånition 2.4. A complex orbifold of complex dimension d is a singular complex space whose singularities are locally isomorphic to quotient singularities C d / G , for Ô¨Ånite sub- groups G ‚äÇ Gl ( d, C ) .\n\nDeÔ¨Ånition 2.5. A diÔ¨Äerential form on a complex orbifold Z is deÔ¨Åned locally at z ‚àà Z as a G -invariant diÔ¨Äerential form on C d where G ‚äÇ Gl ( d, C ) and Z is locally isomorphic to d\n\nRoughly speaking the local geometry of orbifolds reduces to local G -invariant geometry.\n\nWe have a complex of diÔ¨Äerential forms ( A ‚óè ( Z ) , d ) and a double complex ( A ‚óè , ‚óè ( Z ) , ‚àÇ, ¬Ø ‚àÇ ) of bigraded diÔ¨Äerential forms which deÔ¨Åne the de Rham and the Dolbeault cohomology groups (for a Ô¨Åxed p ‚àà N ) respectively:\n\n(1,1)-Lefschetz theorem for projective toric orbifolds\n\nDeÔ¨Ånition 3.1. A subvariety X ‚äÇ P d Œ£ is quasi-smooth if V ( I X ) ‚äÇ A #Œ£ ( 1 ) is smooth outside\n\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub-\n\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub- varieties are quasi-smooth subvarieties (see [2] or [7] for more details).\n\nRemark 3.3 . Quasi-smooth subvarieties are suborbifolds of P d Œ£ in the sense of Satake in [8]. Intuitively speaking they are subvarieties whose only singularities come from the ambient\n\nProof. From the exponential short exact sequence\n\nwe have a long exact sequence in cohomology\n\nH 1 (O ‚àó X ) ‚Üí H 2 ( X, Z ) ‚Üí H 2 (O X ) ‚âÉ 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
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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. ->: the group Hom ( Cl ( Œ£ ) , C ‚àó ) and the group action is induced by the Cl ( Œ£ ) - grading of S .\n\nNow we give a brief introduction to complex orbifolds and we mention the needed theorems for the next section. Namely: de Rham theorem and Dolbeault theorem for complex orbifolds.\n\nDeÔ¨Ånition 2.4. A complex orbifold of complex dimension d is a singular complex space whose singularities are locally isomorphic to quotient singularities C d / G , for Ô¨Ånite sub- groups G ‚äÇ Gl ( d, C ) .\n\nDeÔ¨Ånition 2.5. A diÔ¨Äerential form on a complex orbifold Z is deÔ¨Åned locally at z ‚àà Z as a G -invariant diÔ¨Äerential form on C d where G ‚äÇ Gl ( d, C ) and Z is locally isomorphic to d\n\nRoughly speaking the local geometry of orbifolds reduces to local G -invariant geometry.\n\nWe have a complex of diÔ¨Äerential forms ( A ‚óè ( Z ) , d ) and a double complex ( A ‚óè , ‚óè ( Z ) , ‚àÇ, ¬Ø ‚àÇ ) of bigraded diÔ¨Äerential forms which deÔ¨Åne the de Rham and the Dolbeault cohomology groups (for a Ô¨Åxed p ‚àà N ) respectively:\n\n(1,1)-Lefschetz theorem for projective toric orbifolds\n\nDeÔ¨Ånition 3.1. A subvariety X ‚äÇ P d Œ£ is quasi-smooth if V ( I X ) ‚äÇ A #Œ£ ( 1 ) is smooth outside\n\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub-\n\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub- varieties are quasi-smooth subvarieties (see [2] or [7] for more details).\n\nRemark 3.3 . Quasi-smooth subvarieties are suborbifolds of P d Œ£ in the sense of Satake in [8]. Intuitively speaking they are subvarieties whose only singularities come from the ambient\n\nProof. From the exponential short exact sequence\n\nwe have a long exact sequence in cohomology\n\nH 1 (O ‚àó X ) ‚Üí H 2 ( X, Z ) ‚Üí H 2 (O X ) ‚âÉ 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