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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. 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