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Let $(R,p_1,…,p_n)$ be a Compact Riemann surface of genus $g$ with $n$ marked points. Its deformation space is $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. From Riemann-Roch theorem, it is of dimension $3g-3+n$. Choose $v_1,…v_{3g-3+n}$ supporting away from $p_j, j=1,…n$, which are representatives of a basis of $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. By solving Beltrami equation $$\bar{\partial}f=s_1v_1+…+s_{3g-3+n}v_{3g-3+n}\partial f, |s|<1$$, we can get a new compact surface $\Sigma$ with $n$ marked points from quasi conformal map $f:R\to \Sigma$. Scott. Wolpert says it can be a local chart for Teichuller space.

My question is that what kind of manifold will be obtained from the charts constructed above? Analytic manifold or just a smooth manifold?

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  • $\begingroup$ What do you mean by analytic: Real or complex analytic? $\endgroup$
    – Moishe Kohan
    Commented Jul 25, 2023 at 13:41
  • $\begingroup$ @MoisheKohanonstrike: complex analytic, locally the chart is holomorphic to a polydisk in complex Euclidean space. $\endgroup$
    – CharlieHo
    Commented Jul 25, 2023 at 15:02
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    $\begingroup$ You should add this to the question, as well as the precise reference of Wolpert's paper (he wrote many). Technically speaking, the statement is ill-posed since you have to explain how representatives of $H^1$ are chosen: In order to get complex-analyticity you have to use Beltrami differentials depending holomorphically on $R$. $\endgroup$
    – Moishe Kohan
    Commented Jul 25, 2023 at 15:31

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