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On ${\mathcal{D}}$-modules related to the $b$-function and Hamiltonian flow

Part of: Singularities (Co)homology theory Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  12 October 2018

Thomas Bitoun  and
Travis Schedler
Thomas Bitoun
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK email tbitoun@gmail.com
Travis Schedler
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email trasched@gmail.com
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Abstract

Let $f$ be a quasi-homogeneous polynomial with an isolated singularity in $\mathbf{C}^{n}$. We compute the length of the ${\mathcal{D}}$-modules ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers of $f$ in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When $\unicode[STIX]{x1D706}=-1$ we obtain one more than the reduced genus of the singularity ($\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for $Z$ the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ is nonzero when $\unicode[STIX]{x1D706}$ is a root of the $b$-function of $f$ (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these ${\mathcal{D}}$-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.

Keywords

Hamiltonian flowMilnor numberD-modulesPoisson homologyPoisson varietiesBernstein–Sato polynomialb-functionlocal cohomologynearby cyclesHodge filtrationMilnor fibergenusisolated hypersurface singularities

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 37J05: General theory, relations with symplectic geometry and topology 32S20: Global theory of singularities; cohomological properties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 11 , November 2018 , pp. 2426 - 2440
Copyright
© The Authors 2018 

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Footnotes

The first author was supported by EPSRC grant EP/L005190/1. The second author was partly supported by US National Science Foundation Grant DMS-1406553, and is grateful to the Max Planck Institute for Mathematics in Bonn for excellent working conditions.

References

Alvarez-Montaner, J., Blickle, M. and Lyubeznik, G., Generators of D-modules in positive characteristic , Math. Res. Lett. 12 (2005), 459–473; MR 2155224 (2006m:13024).Google Scholar
Bitoun, T., Length of local cohomology in positive characteristic and ordinarity , Int. Math. Res. Not. IMRN 2018 (2018), doi:10.1093/imrn/rny058.Google Scholar
de Cataldo, M. A. A. and Migliorini, L., The decomposition theorem, perverse sheaves and the topology of algebraic maps , Bull. Amer. Math. Soc. (N.S.) 46 (2009), 535–633; MR 2525735.Google Scholar
Dimca, A., Sheaves in topology, Universitext (Springer, Berlin, 2004); MR 2050072 (2005j:55002).Google Scholar
Etingof, P. and Schedler, T., Poisson traces and D-modules on Poisson varieties , Geom. Funct. Anal. 20 (2010), 958–987.Google Scholar
Etingof, P. and Schedler, T., Invariants of Hamiltonian flow on locally complete intersections , Geom. Funct. Anal. 24 (2014), 1885–1912; MR 3283930.Google Scholar
Etingof, P. and Schedler, T., Coinvariants of Lie algebras of vector fields on algebraic varieties , Asian J. Math. 20 (2016), 795–867; MR 3622317.Google Scholar
Ginsburg, V., Characteristic varieties and vanishing cycles , Invent. Math. 84 (1986), 327–402; MR 833194 (87j:32030).Google Scholar
Kashiwara, M., B-functions and holonomic systems. Rationality of roots of B-functions , Invent. Math. 38 (1976/77), 33–53; MR 0430304 (55 #3309).Google Scholar
Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations , in Algebraic geometry, Tokyo and Kyoto, 1982, Lecture Notes in Mathematics, vol. 1016 (Springer, Berlin, 1983), 134–142; MR 726425 (85e:58137).Google Scholar
Kashiwara, M., D-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217 (American Mathematical Society, Providence, RI, 2003); MR 1943036 (2003i:32018).Google Scholar
Malgrange, B., Polynômes de Bernstein–Sato et cohomologie évanescente , in Analysis and topology on singular spaces, II, III Luminy, 1981, Astérisque, vol. 101 (Société Mathématique de France, Paris, 1983), 243–267; MR 737934 (86f:58148).Google Scholar
Mustaţă, M. and Srinivas, V., Ordinary varieties and the comparison between multiplier ideals and test ideals , Nagoya Math. J. 204 (2011), 125–157; MR 2863367.Google Scholar
Saito, M., $D$ -modules generated by rational powers of holomorphic functions, Preprint (2015), arXiv:1507.01877.Google Scholar
Steenbrink, J., Intersection form for quasi-homogeneous singularities , Compos. Math. 34 (1977), 211–223; MR 0453735 (56 #11995).Google Scholar
Yano, T., On the theory of b-functions , Publ. Res. Inst. Math. Sci. 14 (1978), 111–202; MR 499664.Google Scholar