In the first part, we study the category of D-modules on singular varieties. For an
affine variety X, we show that the derived category of quasi-coherent D-modules is
equivalent to the category of DG modules over a DG algebra, with zeroth cohomology
given by Grothendieck differential operators Diff(X). In the cuspidal case, we recover
Diff(X), relating it to [BN04]. We also compute the cohomology algebra and natural
modules for hypersurfaces, curves, and isolated quotient singularities.
In the second part, we define an analogue of Hochschild homology based on [ES09],
enhancing (zeroth) Poisson homology to a local version via a specific D-module
M(X). Using another D-module Mℏ(X), we introduce Hochschild–de Rham homology,
which generally coincides with Hochschild homology in degree 0 for affine X.
We show that, for certain symplectic resolutions, Poisson-de Rham and Hochschild–
de Rham homology align with de Rham cohomology. Additionally, for X with
finitely many symplectic leaves, Mℏ(X) is holonomic, implying finite generation of
Hochschild–de Rham homology.
In the third part, we apply the second part’s results to the skein theory of tori. We
identify an explicit symplectic resolution Hilb0(T2) of the SLN-character variety of
the 2-torus T2, as a version of the Hilbert scheme of T2. We show that Hilb0(T2)×T2
is a CN×CN cover of Hilb(T2), compute its de Rham cohomology and Hochschild–de Rham homology under quantisation, and deduce dim SkSLN
(T3) = P ⋆ J3(N).