which we assume to be projective. Moreover, T ∶ = N ⊗ Z C ∗ ≃ ( C ∗ ) d is the torus action on P d Σ . We denote by Σ ( i ) the i -dimensional cones\n\nFor a cone σ ∈ Σ, ˆ σ is the set of 1-dimensional cone in Σ that are not contained in σ\n\nand x ˆ σ ∶ = ∏ ρ ∈ ˆ σ x ρ is the associated monomial in S .\n\nDeﬁnition 2.2. The irrelevant ideal of P d Σ is the monomial ideal B Σ ∶ =< x ˆ σ ∣ σ ∈ Σ > and the zero locus Z ( Σ ) ∶ = V ( B Σ ) in the aﬃne space A d ∶ = Spec ( S ) is the irrelevant locus.\n\nProposition 2.3 (Theorem 5.1.11 [5]) . The toric variety P d Σ is a categorical quotient A d ∖ Z ( Σ ) by the group Hom ( Cl ( Σ ) , C ∗ ) and the group action is induced by the Cl ( Σ ) - 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\nDeﬁnition 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 ﬁnite sub- groups G ⊂ Gl ( d, C ) .\n\nDeﬁnition 2.5. A diﬀerential form
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