Katz's middle convolution algorithm

Abstract

This is an expository account of Katz's middle convolution operation on local systems over ${\mathbf{P}}^1-\{q\_1,...,q\_n\}$. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Haraoka-Yokoyama. Kostov's program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and Crawley-Boevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle.

Submitted to Pure and Applied Math Quarterly, Hirzebruch issue. v2 corrects some history and adds references

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Cited by in corpus (1)

• Quivers and difference Painleve equations