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Formal structure of direct image of holonomic \({\mathcal{D}}\)-modules of exponential type. (English) Zbl 1140.32021
If \(\mathcal M\) is a regular holonomic \(\mathcal D_X\)-module on a complex manifold \(X\), we denote by \({\mathcal O}_X[\ast Z]\) the sheaf of meromorphic functions with poles at most along the reduced divisor \(Z\), and let \(g\in H^0({\mathcal O}_X[\ast Z])\) be such a function. Then \({\mathcal M}e^g\) denotes the holonomic \({\mathcal D}_X\)-module obtained by equipping \({\mathcal M}[\ast Z]={\mathcal M}\otimes_{{\mathcal O}_X}{\mathcal O}_X[\ast Z]\) with the connection \(\nabla_g=\nabla +dg\), where \(\nabla\) is the connection on \({\mathcal M}[\ast Z]\) coming from its left \({\mathcal D}_X\)-module structure. Such a holonomic \({\mathcal D}_X\)-module is said to be of exponential type. It is irregular along \(Z\). If \(f\colon X \to C\) is a proper map to a curve then the cohomology sheaves of the direct image of \({\mathcal M}e^g\) under \(f\) are holonomic \({\mathcal D}_C\)-modules but may have irregular singularities.
The purpose of this paper to compute formal invariants of these cohomology modules at their singularities. The problem is thus local on the base, so it is enough to consider \(X=D\times{\mathbb P}^1\) (\(D\) a disc centred at \(0\in{\mathbb C}\)) and \(f=p_1\colon X\to D\), \(g=p_2\colon X\to {\mathbb P}^1\). Then look at \({\mathcal N}_0={\mathcal H}^0 p_{1+}({\mathcal M}e^{p_2})\). Theorem 1 of the paper computes the Newton polygon (and hence the slopes and irregularity number) of \({\mathcal N}_0\) in terms of multiplicities of local irreducible components \(S_\ell\) of the singular support of \({\mathcal N}_0\), and gives a decomposition of the formal irregular part of \({\mathcal N}_0\) after a base change, in terms of Puiseux parametrisations of \(S_\ell\). The characteristic polynomial of the monodromy is computed (Theorem 2) under a genericity assumption. Theorems 3 and 4 reduce other cases to this one at least formally. In particular every formal \({\mathbb C}\big[[t]\big]\langle\partial_t\rangle\)-module is isomorphic after a base change to a germ of a formalised direct image of an analytic \({\mathcal D}\)-module of exponential type.
The main tools are the properties of specialisation and higher direct images established by Y. Laurent and B. Malgrange [Ann. Inst. Fourier 45, No. 5, 1353–1405 (1995; Zbl 0837.35006)], together with local analytic computations in suitable partial resolutions of the singularities.

MSC:
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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