A shortcut in Kapovich’s proof of Haupt’s theorem
The Teichmüller space of genus
curves carries the Hodge bundle
, the total space of which maps into the first cohomology space
via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image
in terms of the integral structure on
and …