Mixed Twistor D-modules

Abstract

We introduce mixed twistor D-modules, and establish the fundamental functorial property. We also prove that they are described as the gluing of admissible variations of mixed twistor structure. In a sense, mixed twistor D-modules could be regarded as a twistor version of mixed Hodge modules due to M. Saito . In the other sense, they could be a mixed version of pure twistor D-modules studied by C. Sabbah and the author. A theory of mixed twistor D-modules is one of the ultimate goals in the study suggested by Simpson's Meta Theorem, and the author hopes that it would play a basic role in the Hodge theory for holonomic D-modules possibly with irregular singularity.Résumé. -2.1.3. Some sheaves. -We recall some sheaves, following Sabbah in [55] and [57], to which we refer for more details and precision. Let X be an n-dimensional complex manifold. Let T be a real C ∞ -manifold. Let V be an open subset ofX×T /T on V with compact supports. Let Diff X×T /T (V ) denote the space of C ∞ -differential operators on V relative to T , i.e., we consider only differentials in the X-direction on V . For any compact subset K ⊂ V and P ∈ Diff X×T /T (V ), we have the semi-norm ϕX×T /T,c (V ), which consists of the sections ϕ such that (P ϕ) | (Z×T )∩V = 0 for any P ∈ Diff X×T /T (V ). We have the induced semi-norms • P,K on the space E <Z,(n,n) X×T /T,c (V ). By the semi-norms, the spaces E (n,n) X×T /T,c (V ) and E <Z,(n,n) X×T /T,c (V ) are locally convex topological spaces. Let C 0 c (T ) denote the space of continuous functions on T with compact supports. It is a normed vector space with the sup norms.The space of continuous C 0 (T )-linear mapsAny elements of Db mod D X×T /T (V ) are called distributions with moderate growth along D. Let Db X×T /T,D (V ) be the space of continuous C ∞ -linear maps Φ :For any compact subset K in V , there exists C K > 0 such that we have the well defined pairingX×T /T,c (V ) with Supp(ϕ) ⊂ K and for Re(s) > C K . It depends on s in a holomorphic way. Let C ∞X×T be the sheaf of C ∞ -functions on X × T . Let D be a hypersurface of X. Let U ⊂ X × T be an open subset. Let ϕ be a C ∞ -function on an open subset U \ (D × T ). We say that ϕ is a C ∞ -function of moderate growth along D, if the following holds for any (Q 1 , Q 2 ) ∈ U ∩ (D × T ).-For any P ∈ Diff X×T /T (U ), there exists N > 0 such that PLet C ∞ mod D X×T denote the sheaf of C ∞ -functions on X × T with moderate growth along D. It is standard to prove that the sheaf is c-soft.Let f : X ′ −→ X be a morphism of complex manifolds. Let D be a hypersurface of X. We put D ′ := f −1 (D).Lemma 2.1.5. -Assume that f is proper and birational. We have a natural isomorphism a 1 : f ! Db mod D ′ X ′ ×T /T ≃ Db mod D X×T /T and an epimorphism a 2 :. We also have f −1 Db mod D

On the Irregular Hodge Filtration of Exponentially Twisted Mixed Hodge modules

“…The theory of mixed twistor D-modules, as developed by Mochizuki [19], is the convenient framework to treat wild Hodge theory. However, this theory produces very few numerical invariants having a Hodge flavor (like Hodge numbers, degrees of Hodge bundles, etc.).…”

“…The proof that we give below uses the full strength of the theory of mixed twistor D-modules of Mochizuki [19].…”

“…(with respect to the z-variable) is part of the data defining an integrable mixed twistor D X ×P 1 -module, according to [19,Proposition 12.5.4].…”

“…The proof of the theorem is given in Section 5.1; it relies much on the theory of mixed twistor D-modules of Mochizuki [19]. This theory allows one to simplify and generalize some of the arguments given in [9], by giving a general framework to treat, from the Hodge point of view, irregular D-modules like E f .…”

“…Let H be a divisor in X , locally defined by a holomorphic function h, and let N be a coherent R X ( * H )-module. According to [19,Definition 3.3.1], one says that N is twistor specializable along H if there exists a coherent R X -submodule N [ * H ] ⊂ N such that, considering locally the graph inclusion i h : X → Y := X × C with the coordinate t on C,…”