Absolute sets of rigid local systems

A set of local systems or vector bundles with an integral connection on a smooth complex algebraic variety is said to be absolute if it satisfies certain compatibility conditions of arithmetic-type with the Riemann-Hilbert correspondence. Absolute sets of rank-one local systems admit particularly nice descriptions, by work of Simpson in the projective case and Budur-Wang in general. For higher rank, there are some conjectures of André-Oort type about the structure of such sets. In this thesis, we will focus on the higher-rank local systems on punctured lines, that is, on hypergeometric local systems. We expect a complete answer to the above conjectures in this case.

Publication year:2022

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