Local deformations of branched projective structures: Schiffer variations and the Teichmüller map (Q2230953)

A complex projective structure, or a \((\operatorname{PSL}_2\mathbb{C},\mathbb{CP}^1)\)-structure, on a surface naturally arises in the context of quasi-Fuchsian manifolds. A branched complex projective structure is a generalization of a complex projective structure that allows branch points. The authors study the structure of the deformation space \(\mathcal{BP}(S)\) of branched complex projective structures on a Riemann surface \(S\) of genus \(g \ge 2\). This deformation space admits the natural forgetful map \(\pi: \mathcal{BP}(S)\to \mathcal{T}(S)\) over the Teichmüller space \(\mathcal{T}(S)\) and the holonomy map \(hol: \mathcal{BP}(S) \to \chi(S)\) over the character variety \(\chi(S)=\operatorname{Hom}(\pi_1(S),\operatorname{PSL}_2\mathbb{C})//\operatorname{PSL}_2\mathbb{C}\). At the same time, \(\mathcal{BP(S)}\) has a stratification in terms of its branch locus. The central object of this paper is the deformation space \(\mathcal{M}_{\lambda,\rho}\subset \mathcal{M}_{k,\rho}\) of branched complex projective structures with a fixed monodromy \(\rho\in \operatorname{Hom}(\pi_1(S),\operatorname{PSL}_2\mathbb{C})\) and fixed branch locus \(\lambda=\{z_1,\dots,z_k\}\subset S\) counted with multiplicity. We assume that \(k\le 2g-2\) and that \(\rho\) is non-elementary. The main result of the paper tells us that if \(\sigma\in \mathcal{M}_{\lambda,\rho}\subset\mathcal{M}_{k,\rho}\) is a critical point for the forgetful map \(\pi^\lambda:\mathcal{M}_{\lambda,\rho}\to \mathcal{T}(S)\) (i.e., the underlying complex structure does not admit any 1st order deformation), then its branching divisor \(div(\sigma)\) is a canonical divisor on the underlying Riemann surface. One should compare this result with the parallel theorem for \(\mathfrak{sl}_2\)-systems which states that the branching divisor for a branched complex projective structure associated with an \(\mathfrak{sl}_2\)-system is always canonical. The authors explicitly find a 1-parameter family of Beltrami differentials \(\mu_t\) along a movement of branch points and compute its 1st order deformation \(\dot{\mu}_0\) at \(t=0\). Since such deformations constitute a neighborhood of \(\sigma\in\mathcal{M}_{\lambda, \rho}\), \(\sigma\) is a critical point for \(\pi^\lambda\) if and only if these 1st order deformations vanish. The authors then argue that, if \(div(\sigma)\) is not canonical, there is a holomorphic quadratic differential \(q\) such that \(\langle q,\dot{\mu}_0\rangle\ne 0\) where \(\langle -,- \rangle\) denotes the natural pairing between holomorphic quadratic differentials and Beltrami differentials. Consequently, \(\sigma\) cannot be a critical point for \(\pi^\lambda\). A similar argument leads us to a partial converse to the above statement in the hyperelliptic setting: a hyperelliptic \(\sigma\in \mathcal{M}_{(1,\dots,1),\rho}\subset \mathcal{M}_{2g-2,\rho}\) is a critical point for \(\pi^{(1,\dots,1)}\) if \(div(\sigma)\) is canonical. Here, a branched complex projective structure \(\sigma\) is hyperelliptic if it admits a projective involution \(J\) with \(2g-2\) fixed points. Under this assumption, \(div(\sigma)\) being canonical guarantees that \(J\) does not fix any branch point of \(\sigma\) and, therefore, one can consider \(J\)-equivariant movements of branch points. The remaining arguments rely on the computation of \(\langle-,\dot{\mu}_0\rangle\) showing that there is a 1-dimensional critical direction passing through \(\sigma\).