Katz's middle convolution algorithm

Details Katz's middle convolution algorithm Katz's middle convolution algorithm

2006-10-17

arxiv.org/abs/math/0610526v2

Mathematics

Algebraic Geometry

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

Scientific paper

This is an expository account of Katz's middle convolution operation on local systems over ${\bf P}^1-\{q\_1,..., q\_n\}$. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following V\"olklein, 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.

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