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Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules

Published online by Cambridge University Press:  24 April 2014

Thomas Reichelt*
Affiliation:
Lehrstuhl VI für Mathematik, Universität Mannheim, Seminargebäude A5, 68131 Mannheim, Germany email thomas.reichelt@math.uni-mannheim.de
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Abstract

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We endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.

Type
Research Article
Copyright
© The Author 2014 

References

Adolphson, A., Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), 269290.CrossRefGoogle Scholar
Adolphson, A. and Sperber, S., A-hypergeometric systems that come from geometry, Proc. Amer. Math. Soc. 140 (2012), 20332042.CrossRefGoogle Scholar
Batyrev, V. V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), 349409.CrossRefGoogle Scholar
Borel, A., Grivel, P.-P., Kaup, B., Haefliger, A., Malgrange, B. and Ehlers, F., Algebraic D-modules, Perspectives in Mathematics, vol. 2 (Academic Press, Boston, MA, 1987).Google Scholar
Brylinski, J.-L., Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque (1986), 3134, 251; Géométrie et analyse microlocales.Google Scholar
D’Agnolo, A. and Eastwood, M., Radon and Fourier transforms for D-modules, Adv. Math. 180 (2003), 452485.CrossRefGoogle Scholar
Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (1990), 255271.CrossRefGoogle Scholar
Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, in Mathematics: Theory & Applications (Birkhäuser, Boston, MA, 1994).Google Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T., D-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236 (Birkhäuser, Boston, MA, 2008); Translated from the 1995 Japanese edition by Takeuchi.CrossRefGoogle Scholar
Kashiwara, M., Quasi-unipotent constructible sheaves, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 757773.Google Scholar
Matusevich, L. F., Miller, E. and Walther, U., Homological methods for hypergeometric families, J. Amer. Math. Soc. 18 (2005), 919941.CrossRefGoogle Scholar
Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005).Google Scholar
Reichelt, T. and Sevenheck, C., Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules, J. Algebraic Geom. (2014), to appear, arXiv:1010.2118 [math.AG].CrossRefGoogle Scholar
Reichelt, T. and Sevenheck, C., Non-affine Landau–Ginzburg models and intersection cohomology, Preprint (2012), arXiv:1210.6527 [math.AG].Google Scholar
Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221333.CrossRefGoogle Scholar
Saito, M., Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323338.CrossRefGoogle Scholar
Saito, M., Primitive ideals of the ring of differential operators on an affine toric variety, Tohoku Math. J. (2) 59 (2007), 119144.CrossRefGoogle Scholar
Stienstra, J., Resonant hypergeometric systems and mirror symmetry, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) (World Scientific, River Edge, NJ, 1998), 412452.Google Scholar
Schmid, W. and Vilonen, K., Hodge theory and unitary representations of reductive Lie groups, in Frontiers of mathematical sciences (International Press, Somerville, MA, 2011), 397420.Google Scholar
Schulze, M. and Walther, U., Irregularity of hypergeometric systems via slopes along coordinate subspaces, Duke Math. J. 142 (2008), 465509.CrossRefGoogle Scholar
Schulze, M. and Walther, U., Hypergeometric D-modules and twisted Gauß–Manin systems, J. Algebra 322 (2009), 33923409.CrossRefGoogle Scholar
Walther, U., Duality and monodromy reducibility of A-hypergeometric systems, Math. Ann. 338 (2007), 5574.CrossRefGoogle Scholar
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