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Cohomology of line bundles: a computational algorithm. (English) Zbl 1314.55012
Summary: 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.{
©2010 American Institute of Physics}

MSC:
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[3] The cohomology computations in Refs. 21-23 were also done using this algorithm.
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[14] Another possibility to check the validity of these results is to perform the “chamber” algorithm described in Chap. 9 of Ref. 13. We have checked that the methods of both algorithms coincide in some simple examples like \(\mathbb{P}^2\) or \(d P_1\), but the computing time is by a factor of approximately \(10^3\) shorter for our algorithm.
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[20] Note that there are a couple of mathematical fine points like the usage of a good cover where all intersections \(U_i \cap U_j\) are contractible. See the mathematical literature for full details.
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