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Register a new userOn D-modules related to the b-function and Hamiltonian flow
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Let f be a quasi-homogeneous polynomial with an isolated singularity. We compute the length of the D-modules $Df^c/Df^{c+1}$ generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. For 1/f we obtain one more than the reduced genus of the singularity. We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the aforementioned quotient is nonzero when c 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 D-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.
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The aim of this Note is to specify the links between the three kinds of Lisbon integrals, trace functions and trace forms with the corresponding D--modules.
We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities, and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein-Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.
On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $mathcal{M}mapstomathcal{M}_{mathrm{reg}}$, called regularization. Recall that $mathcal{M}_{mathrm{reg}}$ is reconstructed from the de Rham complex of $mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.
For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.
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