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 …