Polarizable twistor \(\mathcal D\)-modules.

*(English)*Zbl 1085.32014
Astérisque 300. Paris: Société Mathématique de France (ISBN 2-85629-174-0/pbk). vi, 208 p. (2005).

Let \(\mathcal F\) be a locally constant sheaf of \(\mathbb C\)-vector spaces of finite dimension on a smooth complex projective variety. Let \(f\colon U \to Y\) be a proper holomorphic mapping from an open subset \(U\) of \(X\) in a complex manifold \(Y\).

The main results of this article are the following:

(1) The direct image complex under \(f\) of the associated perverse complex of \(\mathcal F| _U\) decomposes (non-canonically) in a direct sum of its perverse cohomology sheaves.

(2) The relative Hard Lefschetz Theorem holds for these perverse cohomology sheaves.

(3) These perverse cohomology sheaves decompose as the direct sum of intersection complexes supported on closed irreducible analytic subsets of \(Y\).

(4) For \(U = X\) and \(Y\) projective, these perverse cohomology sheaves are semisimple.

(5) For \(Y = \mathbb C\), the graded perverse complexes obtained by grading with respect to the monodromy filtration of the perverse complexes of nearby or vanishing cycles are degree-wise semisimple perverse sheaves on the fibre \(f^{-1}(0).\)

As the author states, one main motivation for the work was M. Kashiwara, conjecture [Semisimple holonomic \({\mathcal D}\)-modules. (Topological field theory, primitive forms and related topics. Proceedings of the 38th Taniguchi symposium, Kyoto, Japan) (1996; Zbl 0935.32009)], which asserts in particular that (1)–(5) holds for any semisimple perverse sheaf with coefficients in \(\mathbb C\) on \(X\), or, more generally, for every semisimple holonomic \(\mathcal D\)-module. This conjecture in turn has been inspired by previous work of A. A. Beilinson, J. Bernstein, P. Deligne [Astérisque 100 (1982; Zbl 0536.14011)] and M. Saito [Publ. Res. Inst. Math. Sci. 26, No. 2, 221–333 (1990; Zbl 0727.14004)], where these results have been obtained for pure perverse sheaves and polarizable Hodge modules, respectively.

One of the main ideas of the proof is the development of the notion of a polarizable twistor \(\mathcal D\)-module, which is central to the article. With this, the author is able to relate the global property of semisimplicity to the variation of some structure, in this case the polarized twistor structure. The ideas of twistor structures are taken from [C. Simpson, Mixed twistor structures (arXiv:math.AG/9705006) (1997)]. The polarizable twistor \(\mathcal D\)-modules are defined by an inductive method (induction on the dimension of the support) that has been inspired by [M. Saito, loc. cit.]. The polarized twistor \(\mathcal D\)-modules are certain triples consisting of two \(\mathcal R_{\mathcal X}\)-modules and a sesquilinear pairing between them into a sheaf of distributions. Here \(\mathcal R_{\mathcal X}\) is a natural sheaf of differential operators on \(\mathcal X := X \times \mathbb C\).

The main results of this article are the following:

(1) The direct image complex under \(f\) of the associated perverse complex of \(\mathcal F| _U\) decomposes (non-canonically) in a direct sum of its perverse cohomology sheaves.

(2) The relative Hard Lefschetz Theorem holds for these perverse cohomology sheaves.

(3) These perverse cohomology sheaves decompose as the direct sum of intersection complexes supported on closed irreducible analytic subsets of \(Y\).

(4) For \(U = X\) and \(Y\) projective, these perverse cohomology sheaves are semisimple.

(5) For \(Y = \mathbb C\), the graded perverse complexes obtained by grading with respect to the monodromy filtration of the perverse complexes of nearby or vanishing cycles are degree-wise semisimple perverse sheaves on the fibre \(f^{-1}(0).\)

As the author states, one main motivation for the work was M. Kashiwara, conjecture [Semisimple holonomic \({\mathcal D}\)-modules. (Topological field theory, primitive forms and related topics. Proceedings of the 38th Taniguchi symposium, Kyoto, Japan) (1996; Zbl 0935.32009)], which asserts in particular that (1)–(5) holds for any semisimple perverse sheaf with coefficients in \(\mathbb C\) on \(X\), or, more generally, for every semisimple holonomic \(\mathcal D\)-module. This conjecture in turn has been inspired by previous work of A. A. Beilinson, J. Bernstein, P. Deligne [Astérisque 100 (1982; Zbl 0536.14011)] and M. Saito [Publ. Res. Inst. Math. Sci. 26, No. 2, 221–333 (1990; Zbl 0727.14004)], where these results have been obtained for pure perverse sheaves and polarizable Hodge modules, respectively.

One of the main ideas of the proof is the development of the notion of a polarizable twistor \(\mathcal D\)-module, which is central to the article. With this, the author is able to relate the global property of semisimplicity to the variation of some structure, in this case the polarized twistor structure. The ideas of twistor structures are taken from [C. Simpson, Mixed twistor structures (arXiv:math.AG/9705006) (1997)]. The polarizable twistor \(\mathcal D\)-modules are defined by an inductive method (induction on the dimension of the support) that has been inspired by [M. Saito, loc. cit.]. The polarized twistor \(\mathcal D\)-modules are certain triples consisting of two \(\mathcal R_{\mathcal X}\)-modules and a sesquilinear pairing between them into a sheaf of distributions. Here \(\mathcal R_{\mathcal X}\) is a natural sheaf of differential operators on \(\mathcal X := X \times \mathbb C\).

Reviewer: Marc Nieper-Wißkirchen (Mainz)

##### MSC:

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |