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Chern connections in the context of CR-geometry

1.Preliminaries: Chern theorem

This post is a result of several discussions with Rodiohn Déev.

Fix a complex manifold X and a complex vector bundle E over X.

Recall that a structure of holomorphic bundle on E is given by an operator

    \[\overline{\partial}_{\mathcal{E}} \colon \Gamma (E) \to \Gamma(E \otimes \Lambda^{0,1}X)\]

which satisfies \overline{\partial}-Leibniz identity

    \[\overline{\partial}_{\mathcal{E}}(fs) = f\overline{\partial}_{\mathcal{E}}(s) + \overline{\partial}(f) \otimes s\]

and the integrability condition

    \[\overline{\partial}_{\mathcal{E}}^2 = 0.\]

For the reader who is more accustomed …

Demystification of the Willmore integrand

The Willmore energy for a surface S in Euclidean 3-space is defined as \tilde{W}(S) = \int_S \mu^2\omega_S, where \mu is the mean curvature of S and \omega_S its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed that the 2-form \Omega_S = (\mu^2-K)\omega_S, where K stands for the Gaussian curvature, is invariant under conformal transformations; its …

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 the intersection pairing \omega \colon \Lambda^2V \to \mathbb{C}. It might be stated as follows.…