Pure Twistor D-module
We recall some of basic definitions in the theory of pure twistor D-modules due
to Sabbah with slightly different notation, for the reference of our later argument.
We also give partial generalization in some part, although they are minor. See 
for more detail and precise.
14.1. 1Z- module
14.1.1. The sheaf 1Z of algebras. Let X be a complex manifold. We put
X = X x CA , where C\ denotes the complex line with the coordinate A. Let
p : X —• X denote the natural projection. Let Ox denote the tangent sheaf of X.
Then we have the subsheaf A • p*@x of P*@x- We denote A • p*@x by @x Since
we do not consider the tangent sheaf of X, there are no confusion.
Let Tx denote the sheaf of differential operators on X. We have the sheaf
p*Vx, which is the sheaf of relative differential operators of X over C. We have
the natural inclusion Ox —• P*Vx- We also have p*@x —• T^x/c- Then we
obtain the inclusion Qx —• l^x/c-
14.1. We obtain the sheaf of subalgebras of p*Tx generated by
Ox and @x- The sheaf is denoted by 1Zx- We will often consider the restriction
of Kx to X* = X xC*. The restriction is denoted by K*x. D
We have the increasing filtration F by the order of differential operators. The
associated graded sheaf is isomorphic to p* Sym' @x-
Let us see the simple case where X has a coordinate (zi,... , zn). We put as
(14.1) di := A • d/dzi.
It is the section of p*Tx/c-
the case, 1Zx is the sheaf of subalgebras of Tx/c,
generated by Ox and 5\ (i = 1,..., n). We have the relation:
In the case, the filtration F is given as follows:
) e Z ^
14.1.2. Left and right 1Z-module. The left 7^-module and the right 11-
module are naturally defined, as in the case of Vx-modules. The category of left
(resp. right) 7^-module on X is denoted by Modl(1Zx) (resp. Modr(1Zx))- We also
have the derived category bounded complex of left (reps, right) 7^-modules which