A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space \(T_g\) of genus \(g\) curves carries the Hodge bundle \(\Omega T_g\), the total space of which maps into the first cohomology space \(V = H^1(S_g,\mathbb{C})\) via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image \(per(\Omega T_g) \subset V\) in terms of the integral structure on \(V = H^1(S, \mathbb{Z}) \otimes \mathbb{C}\) and …