# Cohomology of line bundles: A computational algorithm

@article{Blumenhagen2010CohomologyOL, title={Cohomology of line bundles: A computational algorithm}, author={Ralph Blumenhagen and Benjamin Jurke and Thorsten Rahn and Helmut Roschy}, journal={Journal of Mathematical Physics}, year={2010}, volume={51}, pages={103525-103525} }

We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and type IIB/F-theory compactifications, where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles.

#### 86 Citations

Cohomology of Line Bundles: Proof of the Algorithm

- Mathematics, Physics
- 2010

We present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by R.~Blumenhagen, B.~Jurke and the authors (arXiv:1003.5217) and suggest a… Expand

Computing Cohomology on Toric Varieties

- Mathematics, Physics
- 2011

In these notes a recently developed technique for the computation of line bundle-valued sheaf cohomology group dimensions on toric varieties is reviewed. The key result is a vanishing theorem for the… Expand

Cohomology of line bundles: Applications

- Physics, Mathematics
- 2012

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for… Expand

Computational Tools for Cohomology of Toric Varieties

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- ArXiv
- 2011

Applications to the computation of chiral massless matter spectra in string compactifications are discussed, and using the software package cohomCalg, its utility is highlighted on a new target space dual pair of (0,2) heterotic string models. Expand

Index Formulae for Line Bundle Cohomology on Complex Surfaces

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- 2019

We conjecture and prove closed-form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple… Expand

EULER CHARACTERISTIC OF LINE BUNDLES ON SIMPLICIAL TORICS VIA THE

- 2010

We combine work of Cox on the homogeneous coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for… Expand

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

- Physics, Mathematics
- 2020

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The… Expand

Formulae for Line Bundle Cohomology on Calabi‐Yau Threefolds

- Physics, Mathematics
- Fortschritte der Physik
- 2019

We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi-Yau threefolds realised as complete intersections in products of projective… Expand

EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS VIA THE

- 2010

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for… Expand

Heterotic bundles on Calabi-Yau manifolds with small Picard number

- Physics
- 2011

A bstractWe undertake a systematic scan of vector bundles over spaces from the largest database of known Calabi-Yau three-folds, in the context of heterotic string compactification. Specifically, we… Expand

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