Periodic pencils of flat connections and their p-curvature

A periodic pencil of flat connections on a smooth algebraic variety X is a linear family of flat connections \nabla(s_1,…,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i, where \lbrace x_i\rbrace are local coordinates on X and B_{ij}: X\to {\rm Mat}_N are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts s_j\mapsto s_j+1 up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic p, the p-curvature operators \lbrace C_i,1\le i\le r\rbrace of a periodic pencil \nabla are isospectral to the commuting endomorphisms C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}, where B_{ij}^{(1)} is the Frobenius twist of B_{ij}. This allows us to compute the eigenvalues of the p-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.

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https://harvard.zoom.us/j/96529021352?pwd=ehXEylANVrstFfISgNJhjaPwcIuCby.1