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 [72]
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:
= {(mi,...,ra
) 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
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