Katz's middle convolution algorithm

2006/10/17 by Simpson, Carlos T. · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0610526

Abstract

This is an expository account of Katz's middle convolution operation on local systems over \bf P¹-\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.

Cited by

• Shannon entropy for harmonic metrics on cyclic Higgs bundles II
• Realization of symmetry of A^(1)*₂-surfaces as transformations of logarithmic connections

Related

• On the middle convolution of local systems. With an Appendix by M. Dettweiler and S. Reiter
• Long-Moody construction of braid representations and Katz middle convolution
• On the middle convolution
• Katz's middle convolution and Yokoyama's extending operation
• Galois realizations of classical groups and the middle convolution