Vanishing Cycles and Hermitian Duality

We show the compatibility between the moderate or nearby cycle functor for regular holonomic D-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara. Introduction The Hermitian dual of a D-module was introduced by M. Kashiwara in [9], who showed that the Hermitian dual of a regular holonomic D-module is also regular holonomic (hence coherent). In this paper we show a compatibility result between this functor and the nearby or vanishing cycle functor relative to a holomorphic function for such modules. The latter may be defined using the V -filtration (introduced by Beilinson, Kashiwara and Malgrange). Moreover we make the link with asymptotic expansions of integrals along fibres of the function. This gives a generalization of previous work of D. Barlet on Hermitian duality for the local Gauss-Manin system of an analytic function. In particular this gives a simpler approach to the “tangling phenomenon” described by D. Barlet in [3]. Acknowledgements. This work arose from many discussions with D. Barlet, whom I thank. 1. Hermitian duality 1.a. Notation. Let (X,OX) be a complex analytic manifold of dimension n, (XR,AXR) be the underlying real analytic manifold and let (X,OX = OX) be the complex conjugate manifold. Denote by DX (resp. DX) the sheaf of holomorphic linear differential operators on X (resp. X). Denote by : f 7→ f the R-isomorphism OX → OX and DX → DX . It induces a trivial conjugation functor, sending DX -modules to DX -modules; if M is a DX -module, we denote by M the R-vector space M equipped with the action of DX defined as follows: denote by m the local section m of M viewed as a local section of M ; then P ·m = Pm. Let DbXR (also denoted by DbX for short) be the sheaf of distributions on XR. It acts on the sheaf C-forms φ with compact support of maximal degree, which is a rightDX and DX module. Then DbX is a left DX and DX -module by the formula (PQμ)(φ) = μ(φ·PQ). The sheaf CXR = Db (n,n) X of currents of maximal degree is a right DX and DX -module obtained from DbX by “going from left to right”. It will be convenient in the following to denote by OX,X (resp. DX,X) the sheafOX⊗COX (resp. DX ⊗C DX) and to view DbX (resp. Db (n,n) X ) as a left (resp. right) DX,X -module. 1991 Mathematics Subject Classification. 32S40.