×

zbMATH — the first resource for mathematics

Cohomology of toric line bundles via simplicial Alexander duality. (English) Zbl 1315.55010
Summary: We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B. Jurke et al. [ibid. 51, No. 10, 103525, 15 p. (2010; Zbl 1314.55012)]. We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T. Rahn [ibid. 51, No. 10, 103520, 11 p. (2010; Zbl 1314.55013)], and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on “Serre duality for Betti numbers” which was raised but unresolved in [H. Roschy and T. Rahn, loc. cit.].{
©2011 American Institute of Physics}

MSC:
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
05E45 Combinatorial aspects of simplicial complexes
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Audin, M., The Topology of Torus Actions on Symplectic Manifolds, (1991), Birkhäuser: Birkhäuser, Basel, Boston, Berlin · Zbl 0726.57029
[2] Blumenhagen, R.; Jurke, B.; Rahn, T.; Roschy, H., Cohomology of line bundles: a computational algorithm, J. Math. Phys., 51, 103525, (2010) · Zbl 1314.55012
[3] Björner, A.; Tancer, M., Combinatorial Alexander duality—A short and elementary proof, Discrete Comput. Geom., 42, 4, 586, (2009) · Zbl 1179.57037
[4] Buchstaber, V.; Panov, T., Torus Actions and Their Applications in Topology and Combinatorics, (2002), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 1012.52021
[5] Cox, D., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4, 17, (1995) · Zbl 0846.14032
[6] Cox, D., Little, J., and Schenck, H., Toric Varieties (unpublished), See . · Zbl 1223.14001
[7] Eisenbud, D.; Mustaţǎ, M.; Stillman, M., Cohomology on toric varieties and local cohomology with monomial supports, J. Symb. Comput., 29, 583, (2000) · Zbl 1044.14028
[8] Fulton, W., Introduction to Toric Varieties, (1993), Princeton University Press: Princeton University Press, Princeton, NJ · Zbl 0813.14039
[9] Grünbaum, B., Nerves of simplicial complexes, Aequ. Math., 4, 63, (1970) · Zbl 0193.52903
[10] Maclagan, D.; Smith, G., Multigraded Castelnuovo-Mumford regularity, J. Reine Angew. Math., 571, 179, (2004) · Zbl 1062.13004
[11] Miller, E., The Alexander duality functors and local duality with monomial support, J. Algebra, 231, 1, 180, (2000) · Zbl 0968.13009
[12] Miller, E.; Sturmfels, B., Combinatorial Commutative Algebra, (2005), Springer: Springer, New York · Zbl 1090.13001
[13] Musson, I., Differential operators on toric varieties, J. Pure Appl. Algebra, 95, 303, (1994) · Zbl 0824.14044
[14] Roschy, H.; Rahn, T., Cohomology of line bundles: proof of the algorithm, J. Math. Phys., 51, 103520, (2010) · Zbl 1314.55013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.