Équations différentielles à points singuliers irréguliers en dimension 2

Sabbah, Claude

doi:10.5802/aif.1386

Cited by 67 publications

(140 citation statements)

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“…It follows from Théorème II.2.1.2 in [20], however, that the same proof holds if one has a good formal structure only. The proof relies on an existence theorem for flat solutions to a certain type of partial differential equations whose entries are rapidly decaying as well.…”

Section: Formal Classification and Asymptotic Developmentsmentioning

confidence: 93%

“…We can assume a more general situation, namely any compactification X such that D := X U is normal crossing. Let X denote the real oriented blow-up of D. For the proofs of the various properties stated in the following we refer to [20], II.1.…”

Section: Sheaves Of Functions On the Real Oriented Blow-upmentioning

confidence: 99%

“…Proof: In [21], §7, the assertion about DR <D X is stated assuming a very good formal structure (see the Remark after Théorème 7.2 and 7.3 in [21]). It follows from Théorème II.2.1.2 in [20], however, that the same proof holds if one has a good formal structure only.…”

Section: Formal Classification and Asymptotic Developmentsmentioning

confidence: 99%

“…Additionally, one asks for the elementary model to fulfill a certain technical condition, in which case it is called a good elementary model. In [20], C. Sabbah conjectures that such a good formal decomposition can be achieved after a finite sequence of point blow-ups and cyclic ramification along the smooth strata of the divisor. He proves the conjecture for several classes of connections, in particular it holds if the rank of the connection is lower than or equal to 5.…”

Section: Introductionmentioning

confidence: 99%

“…[20]) indicating a subtle additional feature: If X an denotes an arbitrary compactification of U an with normal crossing divisor D := X an U an as the complement, the desired formal decomposition can be expected after a finite sequence of point blow-ups of X an only. Additionally, one asks for the elementary model to fulfill a certain technical condition, in which case it is called a good elementary model.…”

Section: Introductionmentioning

confidence: 99%

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Periods for irregular singular connections on surfaces

Hien

2006

Math. Ann.

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Given an integrable connection on a smooth quasi-projective algebraic surface U over a subfield k of the complex numbers, we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de Rham cohomology in terms of period integrals. These homology groups generalize the analogous groups in the same situation over curves defined by S. Bloch and H. Esnault. In dimension two, however, new features appear in this context which we explain in detail. Assuming a conjecture of C. Sabbah on the formal classification of meromorphic connections on surfaces (known to be true if the rank is lower than or equal to 5), we prove perfectness of the period pairing in dimension two.

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Section: Formal Classification and Asymptotic Developmentsmentioning

confidence: 93%

Section: Sheaves Of Functions On the Real Oriented Blow-upmentioning

confidence: 99%

Section: Formal Classification and Asymptotic Developmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Periods for irregular singular connections on surfaces

Hien

2006

Math. Ann.

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Irregularity and the period determinant for irregular singular connections on surfaces

Hien

2010

Mathematische Nachrichten

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In [6], we constructed a period pairing for flat irregular singular conncetions on surfaces. We now extend these constructions to a perfect period pairing between the irregularity complex of the connection and the complex of relative rapid decay chains. As a consequence, the period determinant of the connection decomposes as a product of a topological period determinant on the open surface and a determinant coming from the irregularity of the connection. Additionally, we deduce a method to compute the irregularity sheaf up to local isomorphism in topological terms.Periods for irregular singular differential equations occur classically as values of the well-known special functions studied in mathematical physics for instance. The Bessel functions and (confluent) hypergeometric functions together with their integral representations are well-known examples, the integral kernels of these representations being solutions of elementary irregular singular linear differential equations. The functions themselves now solve another irregular singular differential equation, namely the Bessel-or hypergeometric differential equation, which is the Gauß-Manin connection of the original one. A general construction for periods of these type for ordinary systems has been elaborated by S. Bloch and H. Esnault in [1]. They define so-called rapid decay homology groups which pair with the meromorphic de Rham cohomology of the given connection in terms of period integrals.The differential equations for special functions mentioned above also have higher-dimensional generalizations, i.e. partial differential equations, such as the generalized hypergeometric functions and equations after Gelfand and Aomoto (see e.g.[5] for a discussion with a view towards integral representations and periods, the main result however again restricts to the one-dimensional situation).There is a special interest in the determinant of the period pairing (after choices of basis of the meromorphic de Rham cohomology and the rapid decay homology) related via rather mysterious analogies between irregular singular connections as above and wildly ramified -adic sheaves on varieties over finite fields, where the period determinant is related to the ε-factor of the corresponding -adic sheaf.In [6], we started the investigation of the higher-dimensional situation working on complex surfaces. We generalize the construction of Bloch and Esnault to obtain a perfect duality between these and the meromorphic de Rham cohomology assuming a conjecture of C. Sabbah on the formal classification of meromorphic connections in dimension two. This conjecture has now been proven by T. Mochizuki [15], even in aribtrary dimensions [16], for algebraic connections and the duality therefore holds for algebraic connections in arbitrary dimension also (see [7]). More precisely, we consider an algebraic flat connection (E( * D), ∇) on a vector bundle E over the smooth projective variety X with poles along the divisor D. We assume a certain formal structure of ∇ along D (which is possible by a...

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Mixed Twistor D-modules

Mochizuki

2015

Lecture Notes in Mathematics

100

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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 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

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Équations différentielles à points singuliers irréguliers en dimension 2

Cited by 67 publications

References 3 publications

Periods for irregular singular connections on surfaces

Periods for irregular singular connections on surfaces

Irregularity and the period determinant for irregular singular connections on surfaces

Mixed Twistor D-modules

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