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Michael Dettweiler

Publications and source records attributed to Michael Dettweiler.

At least 19 recordsLinked to original sources
arXiv2023

On the $\ell$-adic Fourier transform and the determinant of the middle convolution

We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the étale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz' middle convolution functor $\MC_χ$ in the tame case. The theory of local $ε$-constants then implies that the property of an étale sheaf of having an at most quadratic determinant is often preserved under $\MC_χ$ if $χ$ is quadratic.

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arXiv2023

Monodromy of the Radon transform

We algorithmically determine the monodromy of the local system on the smooth part of the Radon transformation of a generic simple perverse sheaf on the projective plane.

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arXiv2018

Monodromy of elliptic curve convolution, seven-point sheaves of $G_2$-type and motives of Beauville type

We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to $G_2$. This monodromy approach generalizes a result of Katz on the existence of $G_2$-motives in the middle cohomology of deformations of Beauville surfaces.

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arXiv2017

Hodge theory of the middle convolution

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. We give applications of these formulas to local systems with G_2-monodromy.

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arXiv2016

Vector bundles on curves coming from Variation of Hodge Structures

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 \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our negative answer (\cite{cd}) 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 $(\mathbb Z/n)^2$ 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, already considered in joint work of I. Bauer and the first author, 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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arXiv2013

Answer to a question by Fujita on Variation of Hodge Structures

We first provide details for the proof of Fujita's second theorem for Kähler fibre spaces over a curve, asserting that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. Our main result then answers in the negative the question posed by Fujita whether $V$ is semiample. In fact, $V$ is semiample if and only if $Q$ is associated to a representation of the fundamental group of $B$ having finite image. Our examples are based on hypergeometric integrals.

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arXiv2011

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven whose differential Galois group is isomorphic to the exceptional simple algebraic group of type $G_2.$

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arXiv2006

On exceptional rigid local systems

We prove new instances of Simpson's rigidity conjecture which states that quasi-unipotent rigid local systems should be motivic. We construct new relative motives over the fourfold punctured Riemann sphere which give rise to $G_2$-rigid local systems which are not rigid in the group $\GL_7.$

math.AG↗
arXiv2006

Galois realizations of classical groups and the middle convolution

We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of étale cohomology. We derive a formula to compute the topological monodromy of the middle convolution in the general case and use it to deduce some irreducibility criteria. Then we give a geometric interpretation of the middle convolution in the étale setting. This geometric approach to the convolution and the theory of Hecke characters yields information on the occurring arithmetic determinants. We employ these methods to realize special linear groups regularly as Galois groups over ${\bf Q}(t).$

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arXiv2004

Variation of parabolic cohomology and Poincare duality

We continue our study of the variation of parabolic cohomology (math.AG/0310139) and derive an exact formula for the underlying Poincare duality. As an illustration of our methods, we compute the monodromy of the Picard-Euler system and its invariant Hermitian form, reproving a classical theorem of Picard.

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