CMSA Geometry and Physics Seminar: Parabolic de Rham bundles: motivic vs periodic

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July 27, 2020 9:30 pm - 10:30 pm
via Zoom Video Conferencing
Speaker:

Mao Sheng - USTC

Let $C$ be a complex smooth projective curve. We consider the set of parabolic de Rham bundles over $C$ (with rational weights in parabolic structure). Many examples arise from geometry: let $f: X\to U$ be a smooth projective morphism over some nonempty Zariski open subset $U\subset C$. Then the Deligne--Iyer--Simpson canonical parabolic extension of the Gauss--Manin systems associated to $f$ provides such examples. We call a parabolic de Rham bundle \emph{motivic}, if it appears as a direct summand of such an example of geometric origin. It is a deep question in the theory of linear ordinary differential equations and in Hodge theory, to get a characterization of motivic parabolic de Rham bundles. In this talk, I introduce another subcategory of parabolic de Rham bundles, the so-called \emph{periodic} parabolic de Rham bundles. It is based on the work of Lan--Sheng--Zuo on Higgs-de Rham flows, with aim towards linking the Simpson correspondence over the field of complex numbers and the Ogus--Vologodsky correspondence over the finite fields. We show that motivic parabolic de Rham bundles are periodic, and conjecture that they are all periodic parabolic de Rham bundles. The conjecture for rank one case follows from the solution of Grothendieck--Katz p-curvature conjecture, and for some versions of rigid cases should follow from Katz's work on rigid local systems. The conjecture implies that in a spread-out of any complex elliptic curve, there will be infinitely many supersingular primes, a result of N. Elkies for rational elliptic curves. Among other implications of the conjecture, we would like to single out the conjectural arithmetic Simpson correspondence, which asserts that the grading functor is an equivalence of categories from the category of periodic parabolic de Rham bundles to the category of periodic parabolic Higgs bundles. This is a joint work in progress with R. Krishnamoorthy.