Hodge Theory of the Middle Convolution

Dettweiler, Michael; Sabbah, Claude

doi:10.4171/prims/119

Cited by 30 publications

(59 citation statements)

References 22 publications

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“…This construction naturally gives a filtration on V 0 . If we have a VHS F • on E over C, it induces a filtration of every V a simply by taking F p V a := j * F p V ∩ V a this is a well defined vector bundle thanks to Nilpotent orbit theorem (see [DS13]).…”

Section: Deligne Extensionmentioning

confidence: 99%

“…Hypergeometric equations on the sphere are well known to be physically rigid (see [BH89] or [Kat96]) and this rigidity together with irreducibility is enough to endow the flat bundle with a VHS using its associated Higgs bundle structure (see [Fed15] or directly Cor 8.2 in [Sim90]). Using techniques from [Kat96] and [DS13], Fedorov gives in [Fed15] an explicit way to compute the Hodge numbers for the underlying VHS. We extend this computation and give a combinatorial point of view that will be more convenient in the following to express parabolic degrees of the Hodge flag decomposition.…”

Section: Introductionmentioning

confidence: 99%

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Parabolic Degrees and Lyapunov Exponents for Hypergeometric Local Systems

Fougeron

2019

Experimental Mathematics

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Consider the flat bundle on P 1 − {0, 1, ∞} corresponding to solutions of the hypergeometric differential equationFor αi and βj real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on P 1 −{0, 1, ∞}, we associate n Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge structure and study the dependence of the Lyapunov exponents in terms of these degrees by means of numerical simulations. 1 arXiv:1701.08387v1 [math.AG]

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Section: Deligne Extensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parabolic Degrees and Lyapunov Exponents for Hypergeometric Local Systems

Fougeron

2019

Experimental Mathematics

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“…Since the bundle with connection is rigid and irreducible (Propositions 2.4 and 2.2), the existence of CPVHS follows from [Sim2,Cor. 8.1], see also [DS,Thm. 2.4.1].…”

Section: 2mentioning

confidence: 99%

“…1.1].) Then, for a holonomic D-module M on A 1 C , its middle convolution with K α (denoted M C α (M )) is uniquely defined by the condition that its Fourier transform F M C α (M ) is the minimal extension at the origin of F M ⊗ L −α , see [DS,Prop. 1.1.8].…”

Section: Middle Convolutionmentioning

confidence: 99%

Variations of Hodge Structures for Hypergeometric Differential Operators and Parabolic Higgs Bundles

Fedorov

2017

International Mathematics Research Notices

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Abstract. Consider the holomorphic bundle with connection on P 1 −{0, 1, ∞} corresponding to the regular hypergeometric differential operatorIf the numbers α i and β j are real and for all i and j the number α i − β j is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Alessio Corti and Vasily Golyshev. We also use non-abelian Hodge theory to interpret our theorem as a statement about parabolic Higgs bundles.

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“…We remark that the lemma applies in many other contexts, e.g., for more general rigid local systems or motivic local systems whose Hodge numbers can be calculated (cf. [30], [15], [16]). …”

Section: Cyclic Coverings Of the Projective Line Branched Onmentioning

confidence: 99%

Vector bundles on curves coming from variation of Hodge structures

Catanese

Dettweiler

2016

Int. J. Math.

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Abstract. Fujita's second theorem for Kähler fibre spaces over a curve asserts that the direct image V of the relative dualizing sheaf splits as the direct sum V = A ⊕ Q, where A is ample and Q is unitary flat. We focus on our negative answer ([9]) to a question by Fujita: is V semiample?We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group (Z/n) 2 of a Del Pezzo surface of degree 5 (branched on a union of lines forming a bianticanonical divisor), and endowed with a semistable fibration with only 3 singular fibres.The simplest such surfaces are the three ball quotients considered in [3], fibred over a curve of genus 2, and with fibres of genus 4.These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties.

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Hodge Theory of the Middle Convolution

Abstract: Abstract. We compute the behaviour of Hodge data by tensor product with a unitary rank-one local system and middle convolution by a Kummer unitary rank-one local system for an irreducible variation of polarized complex Hodge structure on a punctured complex affine line.

Cited by 30 publications

References 22 publications

Parabolic Degrees and Lyapunov Exponents for Hypergeometric Local Systems

Parabolic Degrees and Lyapunov Exponents for Hypergeometric Local Systems

Variations of Hodge Structures for Hypergeometric Differential Operators and Parabolic Higgs Bundles

Vector bundles on curves coming from variation of Hodge structures

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