Cohomology of line bundles: Proof of the algorithm

Journal of Mathematical Physics

Volumn 51, Issue 10, 2010, Pages

  Roschy, Helmut ^a   Rahn, Thorsten ^a  

^a MAX PLANCK INSTITUT FÜR PHYSIK   (Germany)

Author keywords

[No Author keywords available]

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EID: 78149440424     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3501135     Document Type: Article

References (25)

1. Blumenhagen R. Jurke B. Rahn T. Roschy H. Cohomology of line bundles: A computational algorithm. 2010, and e-print arXiv:cond-mat/1003.5217.
2. The speed-optimized implementation in C++ can be downloaded from and is regularly updated. To get a first experience of the calculations possible, one can also have a quick start with a short Mathematica script that is also available there.
3. Blumenhagen R. Jurke B. Rahn T. Roschy H. Cohomology of line bundles: Applications. and (unpublished).
4. Grayson D. Stillman M. Macaulay 2, a software system for research in algebraic geometry. and Available by ftp at ://.
5. Smith G. In order to do sheaf cohomology computations on general toric varieties, the additional package NORMALTORICVARIETIES.M2 written by is needed. Since this is still work in progress, it is not yet included in the official distribution, but the package content can be copied from his and then separately loaded into MACAULAY2.
6. Cox D.A. Little J.B. Schenck H. Toric Varieties and (unpublished), available at.
7. Cvetic M. Garcia-Etxebarria I. Halverson J. Global F-theory models: Instantons and gauge dynamics. 2010, and e-print arXiv:cond-mat/1003.5337.
8. Note that it can be shown that Čechcohomology on an open cover of a toric variety can be shown to be isomorphic to sheaf cohomology, see Theorem 9.0.4 in Ref. 6.
9. Jow S.-Y. Cohomology of toric line bundles via simplicial Alexander duality. e-print arXiv:cond-mat/1006.0780.
10. In the sense of Chap. 3 of Ref. 6.
11. A condensed introduction to simplicial complexes meeting our requirements is given, e.g by the first chapter of Ref. 15.
12. For a short review of sheaf theory and sheaf cohomology have a look at the appendix of Ref. 1.
13. Note that we always identify Picard group and class group of X, since in the smooth case all Weil divisors are already Cartier.
14. The shift in the rank comes from a shift between the ordinary and the local Čech complex, see also Theorem 9.5.7 in Ref. 6.
15. Miller E. Strumfels B. Combinatorial Commutative Algebra 2005, 227. and Graduate Texts in Mathematics Vol. (Springer, New York, ).
16. Here, the term power set of an ideal stands for taking all possible unions of the generators. In fact, the sequences for remnant cohomology in the algorithm of Ref. 1 come from the combinatorics of this power set and the connection with the full Taylor resolution of S/I will be important for the proof.
17. 3 is among the generators of its Stanley-Reisner ideal, cf. the examples in Ref. 1.
18. See Ref. 25 for more details on these categorical issues.
19. Eisenbud D. Mustaţǎ M. Stillman M. J. Symb. Comput. 2000, 29:583. JSYCEH, 0747-7171, 10.1006/jsco.1999.0326, and.
20. i with i∈σ standing in the denominator. Intuitively, these rationoms can be interpreted as representatives of Čech cohomology on intersections of open sets in the toric variety, cf. Sec. 2.2 of Ref. 1.
21. σ. If one also takes notice of the different combinations of Stanley-Reisner generators that lead to this denominator, one can write down the maps in Eq. and gets a well-defined complex.
22. MacLagan D. Smith G. J. Reine Angew. Math. 2004, 571:179. JRMAA8, 0075-4102, and.
23. Bayer D. Charalambous H. Popescu S. J. Algebra 1999, 221:497. JALGA4, 0021-8693, 10.1006/jabr.1999.7970, and.
24. -1.
25. Weibel C. An Introduction to Homological Algebra 1994, 38. Cambridge Studies in Advanced Mathematics Vol. (Cambridge University Press, Cambridge, England, ).