Raymond Cheng, Pure Math Department, University of Waterloo
“Bundles over Complex Tori”
I shall begin our bundle of fun this semester by discussing holomorphic vector bundles over complex tori. After boring the audience with familiar generalities about vector bundles, I will delve into concrete matters and discuss the problem of classifying holomorphic vector bundles over complex tori. This problem was solved long ago by Atiyah for elliptic curves, elliptic curves being the common alias for complex tori of dimension 1. In higher dimensions, Matsushima and Morimoto have obtained a partial classification for bundles admitting a holo- morphic connection. In presenting these results, I will primarily sketch useful techniques for working with bundles and will hopefully provide some techniques that you can try at home.