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International Mathematics Research Papers (2006) Vol. 2006 : article ID 69590, 116 pages, doi:10.1155/IMRP/2006/69590
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Quiver D-modules and homology of local systems over an arrangement of hyperplanes

S. Khoroshkin and A. Varchenko

Institute for Theoretical and Experimental Physics Moscow 117259, Russia E-mail address: khor{at}itep.ru
Department of Mathematics, University of North Carolina at Chapel Hill NC 27599-3250, USA E-mail address: anv{at}email.unc.edu

Let C be an arrangement of hyperplanes in CN, D the ring of algebraic differential operators on CN. We define a category of quivers associated with C. A quiver is a collection of vector spaces, attached to strata of the arrangement, and suitable linear maps between the spaces . To a quiver we assign a D-module on CN, called a quiver D-module. We describe basic operations for D-modules in terms of linear algebra of quivers. We give an explicit construction of a free resolution of a quiver D-module and use the construction to describe the associated perverse sheaf. As an application, we calculate the cohomology of CN with coefficients in the quiver perverse sheaf (under certain assumptions).



References

  1. Aomoto K. Un théorème du type de Matsushima-Murakami concernant l'intégrale des fonctions multiformes. Journal de Mathématiques Pures et Appliquées. Neuvième Série (1973) 52:1–11.
  2. Arnol'd V. I. The cohomology ring of the colored braid group. Mathematical Notes (1969) 5(2):138–140.
  3. Beilinson A. How to glue perverse sheaves. In: Lecture Notes in Math. (1987) 1289. K-Theory, Arithmetic and Geometry, 1984–1986: Moscow. Berlin: Springer. 42–51.
  4. Beilinson A., Bernstein J., Deligne P. Fasceau perverse. Asterisque (1982) 100.
  5. Bernstein J. Modules over a ring of differential operators. An investigation of the fundamental solutions of equations with constant coefficients. Akademija Nauk SSSR. Funkcional' nyi Analiz i ego Prilozenija (1971) 5(2):1–16.
  6. Bernstein J., Lunts V. Equivariant Sheaves and Functors. In: Lecture Notes in Mathematics (1994) 1578. Berlin: Springer. iv+139.
  7. Borel A., Grivel P.-P., Kaup B., Haefliger A., Malgrange B., Ehlers F. Algebraic D-Modules. In: Perspectives in Mathematics (1987) 2. Massachusetts: Academic Press. xii+355.
  8. Brieskorn E. Sur les groupes de tresses [d'après V. I. Arnol' d]. In: Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401 (1973) 317. Berlin: Springer. 21–44. Lecture Notes in Math.
  9. Gelfand S. I., Manin Y. I. Methods of Homological Algebra (1996) Berlin: Springer. xviii+372.
  10. Goresky M., MacPherson R. Intersection homology. II. Inventiones Mathematicae (1983) 72(1):77–129.[CrossRef][Web of Science]
  11. Grothendieck A. Sur quelques points d'algèbre homologique. The Tohoku Mathematical Journal. Second Series (1957) 9:119–221.
  12. Kashiwara M. B-functions and holonomic systems. Rationality of roots of B-functions. Inventiones Mathematicae (1976–1977) 38(1):33–53.[CrossRef]
  13. Kashiwara M. Vanishing cycle sheaves and holonomic systems of differential equations. In: Lecture Notes in Math. (1983) 1016. Algebraic Geometry, 1982: Tokyo/Kyoto. Berlin: Springer. 134–142.
  14. Kashiwara M. The Riemann-Hilbert problem for holonomic systems. Publications of the Research Institute for Mathematical Sciences, Kyoto University (1984) 20(2):319–365.
  15. Kashiwara M. D-Modules and Microlocal Calculus. In: Translations of Mathematical Monographs (2003) 217. Rhode Island: American Mathematical Society. xvi+254.
  16. Kashiwara M., Kawai T. On holonomic systems of microdifferential equations. III. Systems with regular singularities. Publications of the Research Institute for Mathematical Sciences, Kyoto University (1981) 17(3):813–979.
  17. Khoroshkin S. D-modules over the arrangements of hyperplanes. Communications in Algebra (1995) 23(9):3481–3504.[CrossRef][Web of Science]
  18. Khoroshkin S., Schechtman V. Factorizable D-modules. Mathematical Research Letters (1997) 4(2-3):239–257.[Web of Science]
  19. Khoroshkin S., Schechtman V. Non-resonance D-modules over arrangements of hyperplanes. preprint, 1998, http://arxiv.org/abs/math.AG/9801134.
  20. Malgrange B. Polynômes de Bernstein-Sato et cohomologie évanescente. In: Astérisque (1983) 101-102. Analysis and Topology on Singular Spaces, II, III, 1981: Luminy. Paris: Soc. Math. France. 243–267.
  21. Mebkhout Z. Une équivalence de catégories. Compositio Mathematica (1984) 51(1):51–62.[Web of Science]
  22. Orlik P., Solomon L. Combinatorics and topology of complements of hyperplanes. Inventiones Mathematicae (1980) 56(2):167–189.[CrossRef][Web of Science]
  23. Schechtman V. V., Varchenko A. Arrangements of hyperplanes and Lie algebra homology. Inventiones Mathematicae (1991) 106(1):139–194.[CrossRef][Web of Science]
  24. Varchenko A. Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups. In: Advanced Series in Mathematical Physics (1995) 21. New Jersey: World Scientific. x+371.

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This Article
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