2.5. A diﬀerential form on a complex orbifold Z is deﬁned locally at z ∈ Z as a G -invariant diﬀerential form on C d where G ⊂ 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 diﬀerential forms ( A ● ( Z ) , d ) and a double complex ( A ● , ● ( Z ) , ∂, ¯ ∂ ) of bigraded diﬀerential forms which deﬁne the de Rham and the Dolbeault cohomology groups (for a ﬁxed p ∈ N ) respectively:\n\n(1,1)-Lefschetz theorem for projective toric orbifolds\n\nDeﬁnition 3.1. A subvariety X ⊂ P d Σ is quasi-smooth if V ( I X ) ⊂ A #Σ ( 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 Σ 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 ∗ X ) → H 2 ( X, Z )
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