# 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.

## 97 Citations

### Cohomology of Line Bundles: Proof of the Algorithm

- Mathematics
- 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…

### Computing Cohomology on Toric Varieties

- Mathematics
- 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…

### Cohomology of line bundles: Applications

- 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…

### Computational Tools for Cohomology of Toric Varieties

- MathematicsArXiv
- 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.

### Index Formulae for Line Bundle Cohomology on Complex Surfaces

- MathematicsFortschritte der Physik
- 2020

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…

### EULER CHARACTERISTIC OF LINE BUNDLES ON SIMPLICIAL TORICS VIA THE

- Mathematics
- 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…

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

- 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…

### Formulae for Line Bundle Cohomology on Calabi‐Yau Threefolds

- MathematicsFortschritte 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…

### EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS VIA THE

- Mathematics
- 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…

## References

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### Cohomology of Line Bundles: Proof of the Algorithm

- Mathematics
- 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…

### Cohomology of line bundles: Applications

- 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…

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These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last…

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We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by Blumenhagen, Jurke, Rahn, and Roschy (arXiv:1003.5217). We actually…

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These lecture notes are meant to serve as an introduction to some geometric constructions and techniques (in particular the ones of toric geometry) often employed by the physicist working on string…

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We present and analyze in detail a compact F-theory GUT model in which D-brane instantons generate the top Yukawa coupling non-perturbatively. We elucidate certain aspects of F-theory gauge dynamics…

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The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio…