Katz's middle convolution algorithm
By Carlos Simpson. 2006
53 citations recorded by the source corpus.
This is an expository account of Katz's middle convolution operation on local systems over ${\bf P}^1-\{ q_1,\ldots , q_n\}$. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Volklein, 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.
Subfields: Artificial Intelligence, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b7396bbf9e68ffa94810695f06cca9c410a7e63e
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In Geometry and Topology
Citations in the corpus
- The geometry of maximal representations of surface groups into SO0(2,n) (2017, background): “n SL(n,C)-Higgs bundle (E,Φ) is called cyclic of order k if there is a holomorphic splitting E = E1 ⊕...⊕Ek such that Φ maps Ei into Ei+1 ⊗K (for i < k) and Ek to E1 ⊗K. Proposition 2.31 (Simpson, [Sim09]). If the Higgs bundle (E,Φ) is cyclic of order k, then the cyclic splitting of Φ is orthogon”