Equivariant intersection cohomology of toric varieties

Barthel, Gottfried; Brasselet, Jean‐Paul; Fieseler, Karl-Heinz; Kaup, Ludger

doi:10.1090/conm/241/03627

Cited by 27 publications

(33 citation statements)

References 14 publications

(6 reference statements)

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“…The equivariant cohomology also restricts along open inclusions, but it does not give a sheaf in general (see [BBFK1] for an example). However, it has a sheafification, which is just the sheaf A ∆ of conewise polynomial functions.…”

Section: Theorem 25 ([Kar1]) If ∆ = ∆ P For a Polytope P Then Anymentioning

confidence: 99%

Remarks on the Combinatorial Intersection Cohomology of Fans

Braden¹

2006

Pure and Applied Mathematics Quarterly

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Abstract:We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g 2 . We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that g k (P ) = 0 implies g k (P * ) = 0 and g k+1 (P ) = 0.For a d-dimensional convex polytope P , Stanley [St2] defined a polynomial invariant h(P, t) = d k=0 h k (P )t k of its face lattice, which is usually called the "generalized" or "toric" h-polynomial of P . It is "generalized" in that it extends a previous definition from simplicial polytopes to general polytopes, while the adjective "toric" refers to the fact that if P is a rational polytope (meaning that all its vertices have all coordinates in Q), then the coefficients of h(P, t) are intersection cohomology Betti numbers of an associated projective toric variety X P :(1) h k (P ) = dim R IH 2k (X P ; R).

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Section: Theorem 25 ([Kar1]) If ∆ = ∆ P For a Polytope P Then Anymentioning

confidence: 99%

Remarks on the Combinatorial Intersection Cohomology of Fans

Braden¹

2006

Pure and Applied Mathematics Quarterly

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“…If a simplex τ ∈T corresponds to the simplex τ c ∈T c , then an easy determinant computation shows that V ol(τ c ) = V ol(τ )ck 1 1 · · · ck r r , wherek j = |{i ∈ E j |v i ∈ τ }|.…”

Section: Mixed Residues and Mixed Volumesmentioning

confidence: 99%

Toric residue mirror conjecture for Calabi-Yau complete intersections

Karu

2005

J. Algebraic Geom.

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The toric residue mirror conjecture of Batyrev and Materov [2,3] expresses a toric residue as a power series whose coefficients are certain integrals over moduli spaces. This conjecture for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties was proved independently by Szenes and Vergne [10] and Borisov [5]. We build on the work of these authors to generalize the residue mirror map to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties [3].We start by introducing notation and explaining the main idea of the generalization. We work over the field K = Q. Let M ≃ Z d , let ∆ ⊂ M K be a d-dimensional lattice polytope, and let T be a coherent triangulation of ∆, defined by a convex piecewise linear integral function on ∆. All lattice points in ∆ are assumed to be vertices of the simplices in T . We place ∆ in M K = (M × Z) K as ∆ × {1} and let C ∆ ⊂ M K be the cone over ∆ with vertex 0. Then T defines a subdivision of C ∆ into a fan Σ.The idea of the toric residue mirror conjecture is to relate the semigroup ring S ∆ = K[C ∆ ∩ M] to the cohomology of the fan Σ. Let I ∆ ⊂ S ∆ be the ideal generated by monomials t m where m ∈ M lies in the interior of C ∆ . Given general elements f 0 , . . . , f d ∈ S

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“…On the other hand we consider equivariant cohomology taken with respect to the big torus T = (C * ) n acting on X, [BBFK1]. If X is singular we prefer to replace usual cohomology by the intersection cohomology.…”

Section: Introductionmentioning

confidence: 99%

Weights in the cohomology of toric varieties

Weber

2004

Open Mathematics

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We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex IH * T (X) ⊗ H * (T ). We also describe the weight filtration in IH * (X).

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Equivariant intersection cohomology of toric varieties

Cited by 27 publications

References 14 publications

Remarks on the Combinatorial Intersection Cohomology of Fans

Remarks on the Combinatorial Intersection Cohomology of Fans

Toric residue mirror conjecture for Calabi-Yau complete intersections

Weights in the cohomology of toric varieties

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