"This covers how to load online pdfs into a document format that we can use downstream. This can be used for various online pdf sites such as https://open.umn.edu/opentextbooks/textbooks/ and https://arxiv.org/archive/"
"[Document(page_content='A WEAK ( k, k ) -LEFSCHETZ THEOREM FOR PROJECTIVE TORIC ORBIFOLDS\\n\\nWilliam D. Montoya\\n\\nInstituto de Matem´atica, Estat´ıstica e Computa¸c˜ao Cient´ıfica,\\n\\nFirstly we show a generalization of the ( 1 , 1 ) -Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2 k -dimensional quasi-smooth hyper- surfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational ( k, k ) -cohomology class is algebraic, i.e., the Hodge conjecture holds\\n\\nIn [3] we proved that, under suitable conditions, on a very general codimension s quasi- smooth intersection subvariety X in a projective toric orbifold P d Σ with d + s = 2 ( k + 1 ) the Hodge conjecture holds, that is, every ( p, p ) -cohomology class, under the Poincar´e duality is a rational linear combination of fundamental classes of algebraic subvarieties of X . The proof of the above-mentioned result relies, for p ≠ d + 1 − s , on a Lefschetz\\n\\nKeywords: (1,1)- Lefschetz theorem, Hodge conjecture, toric varieties, complete intersection Email: wmontoya@ime.unicamp.br\\n\\ntheorem ([7]) and the Hard Lefschetz theorem for projective orbifolds ([11]). When p = d + 1 − s the proof relies on the Cayley trick, a trick which associates to X a quasi-smooth hypersurface Y in a projective vector bundle, and the Cayley Proposition (4.3) which gives an isomorphism of some primitive cohomologies (4.2) of X and Y . The Cayley trick, following the philosophy of Mavlyutov in [7], reduces results known for quasi-smooth hypersurfaces to quasi-smooth intersection subvarieties. The idea in this paper goes the other way around, we translate some results for quasi-smooth intersection subvarieties to quasi-smooth hypersurfaces, mainly the ( 1 , 1 ) -Lefschetz theorem.\\n\\nAcknowledgement. I thank Prof. Ugo Bruzzo and Tiago Fonseca for useful discus- sions. I also acknowledge support from FAPESP postdoctoral grant No. 2019/23499-7.\\n\\nPreliminaries and Notation\\n\\nLet M be a free abelian group of rank d , let N = Hom ( M, Z ) , and N R = N ⊗ Z R\\n\\nif there exist k linearly independent primitive elements e\\n\\n, . . . , e k ∈ N such that σ = { µ\\n\\ne\\n\\n+ ⋯ + µ k e k } . • The generators e i are integral if for every i and any nonnegative rational number µ the product µe i is in N only if µ is an integer. • Given two rational simplicial cones σ , σ′ one says that σ′ is a face of σ ( σ′ < σ ) if the set of integral generators of σ′ is a subset of the set of integral generators of σ . • A finite set Σ = { σ\\n\\n, . . . , σ t } of rational simplicial cones is called a rational simplicial complete d -dimensional fan if:\\n\\nall faces of cones in Σ are in Σ ;\\n\\nif σ, σ′ ∈ Σ then σ ∩ σ′ < σ and σ ∩ σ′ < σ′ ;\\n\\nN R = σ\\n\\n∪ ⋅ ⋅ ⋅ ∪σ t .\\n\\nA rational simplicial complete d -dimensional fan Σ defines a d -dimensional toric variety P d Σ having only orbifold singularities 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, ˆσ is the set of 1-dimensional cone in Σ that are not contained in σ\\n\\nand x ˆσ∶ = ∏ ρ ∈ ˆσ 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 ˆσ∣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 sin
