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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/

from langchain.document_loaders import OnlinePDFLoader

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[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, 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 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 ) → H 2 (O X ) ≃ H 0 , 2 ( X )\n\nwhere the last isomorphisms is due to Steenbrink in [9]. Now,\n\nH 2 ( X, Z ) / / (cid:15) (cid:15) H 2 ( X, O X ) ≃ Dolbeault (cid:15) (cid:15) H 2 ( X, C ) deRham ≃ (cid:15) (cid:15) H 2 dR ( X, C ) / / H 0 , 2 ¯ ∂ ( X )\n\nof the proof follows as the ( 1 , 1 ) -Lefschetz theorem in [6].\n\nRemark 3.5 . For k = 1 and P d Σ 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\n\ngiven by the Lefschetz morphism and since it is a morphism of Hodge structures, we have:\n\nH 1 , 1 ( X, Q ) ≃ H dim X − 1 , dim X − 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 Σ and let π ∶ P ( E ) → P d Σ be the projective space bundle associated to the vector bundle E = L 1 ⊕ ⋯ ⊕ L s . It is known that P ( E ) is a ( d + s − 1 ) -dimensional simplicial toric variety whose fan depends on the degrees of the line bundles and the fan Σ. Furthermore, if the Cox ring, without considering the grading, of P d Σ is C [ x 1 , . . . , x m ] then the Cox ring of P ( E ) is\n\nMoreover for X a quasi-smooth intersection subvariety cut oﬀ by f 1 , . . . , f s with deg ( f i ) = [ L i ] we relate the hypersurface Y cut oﬀ by F = y 1 f 1 + ⋅ ⋅ ⋅ + 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 − 1 Σ ,X to keep track of its relation with X and P d Σ .\n\nThe following is a key remark.\n\nRemark 4.1 . There is a morphism ι ∶ X → Y ⊂ P d + s − 1 Σ ,X . Moreover every point z ∶ = ( x, y ) ∈ Y with y ≠ 0 has a preimage. Hence for any subvariety W = V ( I W ) ⊂ X ⊂ P d Σ there exists W ′ ⊂ Y ⊂ P d + s − 1 Σ ,X such that π ( W ′ ) = W , i.e., W ′ = { z = ( x, y ) ∣ x ∈ W } .\n\nFor X ⊂ P d Σ a quasi-smooth intersection variety the morphism in cohomology induced by the inclusion i ∗ ∶ H d − s ( P d Σ , C ) → H d − s ( X, C ) is injective by Proposition 1.4 in [7].\n\nDeﬁnition 4.2. The primitive cohomology of H d − s prim ( X ) is the quotient H d − s ( X, C )/ i ∗ ( H d − s ( P d Σ , C )) and H d − s prim ( X, Q ) with rational coeﬃcients.\n\nH d − s ( P d Σ , C ) and H d − s ( X, C ) have pure Hodge structures, and the morphism i ∗ is com- patible with them, so that H d − s prim ( X ) gets a pure Hodge structure.\n\nThe next Proposition is the Cayley proposition.\n\nProposition 4.3. [Proposition 2.3 in [3] ] Let X = X 1 ∩⋅ ⋅ ⋅∩ X s be a quasi-smooth intersec- tion subvariety in P d Σ cut oﬀ by homogeneous polynomials f 1 . . . f s . Then for p ≠ d + s − 1 2 , d + s − 3 2\n\nRemark 4.5 . The above isomorphisms are also true with rational coeﬃcients since H ● ( X, C ) = H ● ( X, Q ) ⊗ Q C . See the beginning of Section 7.1 in [10] for more details.\n\nTheorem 5.1. Let Y = { F = y 1 f 1 + ⋯ + y k f k = 0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to the quasi-smooth intersection surface X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f k ⊂ P k + 2 Σ . Then on Y the Hodge conjecture holds.\n\nthe Hodge conjecture holds.\n\nProof. If H k,k prim ( X, Q ) = 0 we are done. So let us assume H k,k prim ( X, Q ) ≠ 0. By the Cayley proposition H k,k prim ( Y, Q ) ≃ H 1 , 1 prim ( X, Q ) and by the ( 1 , 1 ) -Lefschetz theorem for projective\n\ntoric orbifolds there is a non-zero algebraic basis λ C 1 , . . . , λ C n with rational coeﬃcients of H 1 , 1 prim ( X, Q ) , that is, there are n ∶ = h 1 , 1 prim ( X, Q ) algebraic curves C 1 , . . . , C n in X such that under the Poincar´e duality the class in homology [ C i ] goes to λ C i , [ C i ] ↦ λ C i . Recall that the Cox ring of P k + 2 is contained in the Cox ring of P 2 k + 1 Σ ,X without considering the grading. Considering the grading we have that if α ∈ Cl ( P k + 2 Σ ) then ( α, 0 ) ∈ Cl ( P 2 k + 1 Σ ,X ) . So the polynomials deﬁning C i ⊂ P k + 2 Σ can be interpreted in P 2 k + 1 X, Σ but with diﬀerent degree. Moreover, by Remark 4.1 each C i is contained in Y = { F = y 1 f 1 + ⋯ + y k f k = 0 } and\n\nfurthermore it has codimension k .\n\nClaim: { C i } ni = 1 is a basis of prim ( ) . It is enough to prove that λ C i is diﬀerent from zero in H k,k prim ( Y, Q ) or equivalently that the cohomology classes { λ C i } ni = 1 do not come from the ambient space. By contradiction, let us assume that there exists a j and C ⊂ P 2 k + 1 Σ ,X such that λ C ∈ H k,k ( P 2 k + 1 Σ ,X , Q ) with i ∗ ( λ C ) = λ C j or in terms of homology there exists a ( k + 2 ) -dimensional algebraic subvariety V ⊂ P 2 k + 1 Σ ,X such that V ∩ Y = C j so they are equal as a homology class of P 2 k + 1 Σ ,X ,i.e., [ V ∩ Y ] = [ C j ] . It is easy to check that π ( V ) ∩ X = C j as a subvariety of P k + 2 Σ where π ∶ ( x, y ) ↦ x . Hence [ π ( V ) ∩ X ] = [ C j ] which is equivalent to say that λ C j comes from P k + 2 Σ which contradicts the choice of [ C j ] .\n\nRemark 5.2 . Into the proof of the previous theorem, the key fact was that on X the Hodge conjecture holds and we translate it to Y by contradiction. So, using an analogous argument we have:\n\nargument we have:\n\nProposition 5.3. Let Y = { F = y 1 f s +⋯+ y s f s = 0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to a quasi-smooth intersection subvariety X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f s ⊂ P d Σ such that d + s = 2 ( k + 1 ) . If the Hodge conjecture holds on X then it holds as well on Y .\n\nCorollary 5.4. If the dimension of Y is 2 s − 1 , 2 s or 2 s + 1 then the Hodge conjecture holds on Y .\n\nProof. By Proposition 5.3 and Corollary 3.6.\n\n[\n\n] Angella, D. Cohomologies of certain orbifolds. Journal of Geometry and Physics\n\n(\n\n),\n\n–\n\n[\n\n] Batyrev, V. V., and Cox, D. A. On the Hodge structure of projective hypersur- faces in toric varieties. Duke Mathematical Journal\n\n,\n\n(Aug\n\n). [\n\n] Bruzzo, U., and Montoya, W. On the Hodge conjecture for quasi-smooth in- tersections in toric varieties. S˜ao Paulo J. Math. Sci. Special Section: Geometry in Algebra and Algebra in Geometry (\n\n). [\n\n] Caramello Jr, F. C. Introduction to orbifolds. a\n\niv:\n\nv\n\n(\n\n). [\n\n] Cox, D., Little, J., and Schenck, H. Toric varieties, vol.\n\nAmerican Math- ematical Soc.,\n\n[\n\n] Griffiths, P., and Harris, J. Principles of Algebraic Geometry. John Wiley & Sons, Ltd,\n\n[\n\n] Mavlyutov, A. R. Cohomology of complete intersections in toric varieties. Pub- lished in Paciﬁc J. of Math.\n\nNo.\n\n(\n\n),\n\n–\n\n[\n\n] Satake, I. On a Generalization of the Notion of Manifold. Proceedings of the National Academy of Sciences of the United States of America\n\n,\n\n(\n\n),\n\n–\n\n[\n\n] Steenbrink, J. H. M. Intersection form for quasi-homogeneous singularities. Com- positio Mathematica\n\n,\n\n(\n\n),\n\n–\n\n[\n\n] Voisin, C. Hodge Theory and Complex Algebraic Geometry I, vol.\n\nof Cambridge Studies in Advanced Mathematics . Cambridge University Press,\n\n[\n\n] Wang, Z. Z., and Zaffran, D. A remark on the Hard Lefschetz theorem for K¨ahler orbifolds. Proceedings of the American Mathematical Society\n\n,\n\n(Aug\n\n).\n\n[2] Batyrev, V. V., and Cox, D. A. On the Hodge structure of projective hypersur- faces in toric varieties. Duke Mathematical Journal 75, 2 (Aug 1994).\n\n[\n\n] Bruzzo, U., and Montoya, W. On the Hodge conjecture for quasi-smooth in- tersections in toric varieties. S˜ao Paulo J. Math. Sci. Special Section: Geometry in Algebra and Algebra in Geometry (\n\n).\n\n[3] Bruzzo, U., and Montoya, W. On the Hodge conjecture for quasi-smooth in- tersections in toric varieties. S˜ao Paulo J. Math. Sci. Special Section: Geometry in Algebra and Algebra in Geometry (2021).\n\nCaramello Jr, F. C. Introduction to orbifolds. arXiv:1909.08699v6 (2019).\n\nA. R. Cohomology of complete intersections in toric varieties. Pub-', lookup_str='', metadata={'source': '/var/folders/bm/ylzhm36n075cslb9fvvbgq640000gn/T/tmpzh8ofn_m/online_file.pdf'}, lookup_index=0)]