Higgs Bundle in Geometry and Arithmetic A Proposal

Abstract: The notion of a Higgs bundle originated from the theory of elementary particles, more precisely from the notion of the Higgs boson (field) in particle physics. The Yukawa coupling, named after Hideki Yukawa, is used in the Standard Model in particle physics to describe the coupling between the Higgs bosons, a.k.a. the God particle, and the quarks and leptons. A major development in complex nonabelian Hodge theory was made by Hitchin, Donadlson, Uhlenbeck-Yau and Simpson in the so-called Hitchin-DonaldsonUhlenbeck-Yau-Simpson correspondence, a powerful tool in complex algebraic/analytic geometry. In this talk I will raise a proposal for exploring , exploiting and extending further our newly developed theories of Higgs bundles in algebraic and arithmetic geometry. We will focus principally on the following two programs: • The Shafarevich Program: We work on moduli spaces of polarized varieties in our approach to (1) the Shafarevich conjecture on the finiteness of isomorphism classes of families of higher dimensional varieties and (2) a folklore conjecture on the bigness of the fundamental group of moduli spaces. • p -adic Nonabelian Hodge Theory: We develop and explore further a theory of Higgs bundles on varieties over p-adic fields. Three directions of applications are (1) to Faltings p-adic Simpson correspondence and its relation to Scholze’s OBdR-functor, (2) revisiting Grothendieck anabelian geometry via nonabelian Hodge-Tate comparison and (3) to the construction of motivic local systems over p-adic curves in connections to Drinfeld’s work on the Langlands program via Abe’s solution of Deligne’s conjecture on p to ` companions. The proposal will therefore demonstrate that the concept of Higgs bundle in various generalized settings plays a fundamental role in connecting different fields in algebraic geometry and topology via Yukawacoupling and in arithmetic geomety via p-adic Higgs de Rham flow, a p-adic analogue of Yang-Mills-Higgs equation over the archimadean field. Remarkably the both notions originally came from particle physics and String theory via Calabi-Yau manifolds. I have discussed with Steven Lu, Ruiran Sun and Jinbang Yang on various parts of the proposal. I thank them very much.

Mathematics

Audience: researchers in the topic

ICCM 2020