On the cohomology of a simple normal crossings divisor

Bakhtary, Parsa

doi:10.1016/j.jalgebra.2010.07.023

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…Note added: after this work was completed and posted as a preprint, we became aware of [16], which arrives at results that have some overlap with ours, in the special case that the CW complex is in fact a simplicial complex.…”

Section: Introductionmentioning

confidence: 92%

The Hodge Numbers of Divisors of Calabi‐Yau Threefold Hypersurfaces

Braun

Long

McAllister

et al. 2020

Fortschritte der Physik

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We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor D requires counting fermion zero modes, which depend on the Hodge numbers h i ( D). Suppose that X is a smooth Calabi-Yau threefold hypersurface in a toric variety V, and let D be the restriction to X of a square-free divisor of V. We give a formula for h i ( D) in terms of combinatorial data. Moreover, we construct a CW complex D such that h i ( D) = h i (D). We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large h 1,1. As an illustration we compute the Hodge numbers of a class of divisors in a threefold with h 1,1 = 491. Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi-Yau hypersurfaces.

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Section: Introductionmentioning

confidence: 92%

The Hodge Numbers of Divisors of Calabi‐Yau Threefold Hypersurfaces

Braun

Long

McAllister

et al. 2020

Fortschritte der Physik

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“…. Furthermore, by Remark 5.5 in [4], H i (O D ) depends entirely on the simplicial complex ∆(D), which is constructed using only the incidence information among the D i 0 ...ip .…”

Section: A Technical Lemmamentioning

confidence: 99%

Asymptotic syzygies of normal crossing varieties

Chun¹

2017

Preprint

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Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. In that case, we compute the asymptotic syzygies in terms of the cohomology of the simplicial complex associated to the normal crossing variety.We combine our results on normal crossing varieties with knowledge of degenerations of certain smooth projective varieties to obtain some results on asymptotic syzygies of those smooth projective varieties.

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Weights on cohomology, invariants of singularities, and dual complexes

2013

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Abstract. We study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size, when it is the exceptional divisor of a singularity. We also give bounds for the cohomology of links. The invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities.Given a divisor on a variety, the combinatorics governing the way the components intersect is encoded by the associated dual complex. This is the simplicial complex with p-simplices corresponding to (p + 1)-fold intersections of components of the divisor. Kontsevich and Soibelman [KS, A.4] and Stepanov [Stp1] had independently observed that the homotopy type of the dual complex of a simple normal crossing exceptional divisor associated to a resolution of an isolated singularity is an invariant for the singularity. In fact, [KS], and later Thuillier [T] and Payne [P] have obtained homotopy invariance results for more general dual complexes, such as those arising from boundary divisors. In characteristic zero, all these results are consequences of the weak factorization theorem of W lodarczyk [Wlo], and Abramovich-Karu-Matsuki-W lodarczyk [AKMW]; Thuillier uses rather different methods based on Berkovich's non-Archimedean analytic geometry. As we show here, a slight refinement of factorization (theorems 7.6, 7.7) and of these techniques yields some generalizations this statement. This applies to divisors of resolutions of arbitrary not necessarily isolated singularities, and even in a more general context (discussed in the final section). We also allow dual complexes associated with nondivisorial varieties, such as fibres of resolutions of nonisolated singularites. Here is a slightly imprecise formulation of theorems 7.5 and 7.9. [KS], although we were unaware of this paper at the time these results were completed).To put the remaining results in context, we note that the present paper can be considered as the extended version of our earlier preprint [ABW], where we gave

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On the cohomology of a simple normal crossings divisor

Cited by 3 publications

References 23 publications

The Hodge Numbers of Divisors of Calabi‐Yau Threefold Hypersurfaces

The Hodge Numbers of Divisors of Calabi‐Yau Threefold Hypersurfaces

Asymptotic syzygies of normal crossing varieties

Weights on cohomology, invariants of singularities, and dual complexes

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