In this paper we present a new and elementary approach for proving the main results of Katz (1996) using the Jordan-Pochhammer matrices of Takano and Bannai (1976) and Haraoka (1994). We find an explicit version of the middle convolution of Katz (1996) that connects certain tuples of matrices in linear groups. From this, Katz' existence algorithm for rigid tuples in linear groups can easily be deduced. It can further be shown that the convolution operation on tuples commutes with the braid group action.This yields a new approach in inverse Galois theory for realizing subgroups of linear groups regularly as Galois groups over Q. This approach is then applied to realize numerous series of classical groups regularly as Galois groups over Q.In the Appendix we treat an additive version of the convolution.

In [M. Dettweiler, S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000) 761-798], a purely algebraic analogon of Katz' middle convolution functor (see [N.M. Katz, Rigid Local Systems, Ann. of Math. Stud., vol. 139, Princeton University Press, 1997])is given. In this paper, we find an explicit Riemann-Hilbert correspondence for this functor. This leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the Riemann-Hilbert correspondence.

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.

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.

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