Lakeside Lectures

Speaker: Professor Yoichi Miyaoka (The University of Tokyo)
Title:  Vector bundles associated with Higgs bundles with applications to the Bogomolov inequality of semistable Higgs bundles
Abstract: A Higgs bundle on a smooth variety  is a pair  of a vector bundle and a homomorphism . If a subsheaf of  satisfies , it is called a Higgs subsheaf. When  is projective with an ample divisor H, we can define the semistability of a Higgs bundle  in the same manner as in the case of usual vector bundles. In the lecture, we show that a Higgs bundle  gives rise to a vector bundle  defined over a variety  together with a surjective morphism  such that (1)  has the same rank as ; (2) Higgs subsheaves  of  corresponds to a subsheaf of ; and (3) Semistability of  translates to a weak semistability of . From the above correspondence between  and , we deduce several results such as: (1) Semistability is preserved by pull-backs and tensor products; (2) The 1st and the 2nd Chern classes of a semistable Higgs bundle satisfy the Bogomolov inequality. The Bogomolov inequality was first proved by Simpson by constructing harmonic metrics on Higgs bundles, which depends on hard analysis. Our approach gives a much more elementary alternative proof.
Date:  Dec. 9th, 2013
Time: 14:00-15:00
Venue: Room 202,2/F, Astro-Math. Building
Refreshment:13:30-14:00
Organizers: Yi-Chiuan Chen, Chen-Yu Chi, Chun-Chung Hsieh, Jeng-Daw Yu