BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Pengfei Huang (Universität Heidelberg) DTSTART:20210521T060000Z DTEND:20210521T080000Z DTSTAMP:20260423T035720Z UID:WiSe/19 DESCRIPTION:Title: Wh at is nonabelian Hodge theory?\nby Pengfei Huang (Universität Heidelb erg) as part of What is ...? Seminar\n\n\nAbstract\nNonabelian Hodge theor y can be thought as nonabelian analogue of (abelian) Hodge theory by repla cing the abelian (coefficient) groups into nonabelian (coefficient) groups . This is mainly due to the celebrated work of Donaldson\, Corlette\, Hitc hin\, and Simpson\, which gives us a correspondence between local systems and Higgs bundles. More precisely\, the nonabelian Hodge theory gives an e quivalence between the category of reductive representations of the fundam ental group\, the category of semisimple flat bundles\, and the category o f polystable Higgs bundles with vanishing rational Chern classes\, through pluri-harmonic metrics. Moreover\, such an equivalence of categories is f unctorial\, and preserves tensor products\, direct sums\, and duals. In mo duli viewpoint\, this theory indicates that\, the moduli space of irreduci ble representations (called character variety\, or Betti moduli space)\, as a smooth affine variety\, is complex analytic isomorphic to the moduli space of irreducible flat bundles (called de Rham moduli space)\, which is a smooth Stein manifold (in the sense of analytic topology)\, and is real analytic isomorphic to the moduli space of stable Higgs bundles (called D olbeault moduli space)\, which is a smooth quasi-projective variety. All o f these objects can be generalized to a family of flat λ-connections para metrized by λ ∈ C\, a notion introduced by Deligne\, further studied by Simpson\, and Mochizuki. \n\nIn this talk\, I will begin with a quick rev iew of (abelian) Hodge theory as the motivation of this theory. Then I wil l introduce this theory precisely from an analytic viewpoint by introducin g the work of Donaldson\, Corlette\, Hitchin\, Simpson\, and Mochizuki on the existence of pluri-harmonic metrics. Then I will talk about this theor y from the moduli viewpoint. A good reference of this theory is a survey p aper by S. Rayan and A. Garcı́a-Raboso ( “Introduction to nonabelian H odge theory: flat connections\, Higgs bundles\, and complex variations of Hodge structure\, Fields Inst. Monogr. 34 (2015)\, 131-171.”)\, you can also take the first chapter of my thesis as a reference.\n LOCATION:https://researchseminars.org/talk/WiSe/19/ END:VEVENT END:VCALENDAR