The algorithm for the computation of sheaf cohomologies for line bundles on toric varieties presented in arXiv:1003.5217 [hep-th] "Cohomology of Line Bundles: A Computational Algorithm" has been implemented in a convenient and high-performance C/C++ application called cohomCalg, which is available for download on this page and subject to future updates and/or optimizations. The optional cohomCalg Koszul extension serves as a Mathematica 7 frontend and allows for the easy computation of hypersurface and complete intersection cohomologies, following the material presented in arXiv:1010.3717 [hep-th].
About three months after the initial conjecture's preprint release, a proof of the algorithm was presented in arXiv:1006.2392 [hep-th], which also clarifies much of the underlying mathematical structures. At the same time an independant proof was developed in arXiv:1006.0780 [math.AG] - published in fact a few days earlier - which utilized alternative methods.
The script package is a regular ZIP archive, which can be opened and/or extracted using basically any standard application for archive files. In the Linux/Unix console, simply use the "unzip" command. The source code is freely available under the GNU GPL v3 license terms. Furthermore, the implementation makes use of The Polyhedral Library (or PolyLib for short), which is available here under the same license.
Documentation:A full documentation and several example input files are included in the cohomCalg download package, see ►"manual.pdf". Furthermore, you can eMail us for technical support or other related questions. For ease of use the package includes pre-compiled binaries for Microsoft Windows both in 32- and 64-bit versions, with the latter being considerably faster.
Certain invalid input data is at the moment not safely handled by the PolyLib. For the moment, the crash situation is circumvented in a Quick&Dirty manner. In such cases you will see a message like "Counting of the rationoms errorneous - is your input geometry valid?" in those cases, which caused the original release version to crash. So far, all such cases could be traced back to bad input data. Further implementation-related shortcomings are explained in the manual.
Aside from bug fixes, nothing is scheduled at the moment.
As mentioned the entire package is published under the GNU GPL v3 License, as required by the included PolyLib. This means that any derivative work also has to be published under the GPL v3 License or an equivalent license. The download package contains a small text file "Proper Citation.txt" providing a BibTeX entry for cohomCalg, which you can use if you use the program in your work. Or you can simply Copy&Paste the following BibTeX snippet provided for convenience:
@Article{Blumenhagen:2010pv,
author = "Blumenhagen, Ralph and Jurke, Benjamin
and Rahn, Thorsten and Roschy, Helmut",
title = "{Cohomology of Line Bundles: A Computational Algorithm}",
journal = "J. Math. Phys.",
volume = "51",
pages = "103525",
issue = "10",
year = "2010",
doi = "10.1063/1.3501132",
eprint = "1003.5217",
archivePrefix = "arXiv",
primaryClass = "hep-th"}
@Misc{cohomCalg:Implementation,
title = "{cohomCalg package}",
howpublished = "Download link",
url = "http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg/",
note = "High-performance line bundle cohomology computation based on \cite{Blumenhagen:2010pv}",
year = "2010"}
In order to derive the Stanley-Reisner ideal, which is a required input for the program, you may want to take a look at TOPCOM, which can also enumerate all possible fans for a given set of vertices. The Maple script package SCHUBERT can be used to compute intersection numbers and further geometrical quantities of toric varieties. Furthermore, there is the package PALP which is useful for computing invariants of hypersurfaces, Mori cone vectors etc. You may also want to take a look at the SAGE Library of freely available mathematical software. The Macaulay2 software, which allows similar computations, was heavily used during the development process.
A couple of questions you may have regarding this project and the cohomCalg implementation are answered in a F.A.Q.
COPYRIGHT INFO: The lightbulb images used in the application logo are made by DragonArt and are used under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 license.
The three icons used for the link arrow, changelog and bug listing were made by DryIcons and are used under the DryIcons Free License Agreement.
Mathematica is a registered trademark of Wolfram Research, Inc. All rights reserved. Windows is a registered trademark of Microsoft Corporation in the United States and other countries.
Website Design & Maintenance: Benjamin Jurke
