Uniformization of branched surfaces and Higgs bundles

8 Jan 2021  ·  Indranil Biswas, Steven Bradlow, Sorin Dumitrescu, Sebastian Heller ·

Given a compact Riemann surface $\Sigma$ of genus $g_\Sigma\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $\Sigma$ with $\text{degree}(D)\, <\, 2(g_\Sigma -1)$, there is a unique cone metric on $\Sigma$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2\pi n_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $\Sigma$ parametrized by a nonempty open subset of $H^0(\Sigma,\,K_\Sigma^{\otimes 2}\otimes{\mathcal O}_\Sigma(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [Hi1], for the case $D\,=\, 0$.

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Differential Geometry Algebraic Geometry