Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Abstract

Let X denote a noncompact finite volume hyperbolic Riemann surface of genus \(g\ge 2\), with only one puncture at \(i\infty \) (identifying X with its universal cover \({\mathbb {H}}\)). Let \({{{\overline{X}}}}:=X\cup \lbrace i\infty \rbrace \) denote the Satake compactification of X. Let \(\Omega _{{{{\overline{X}}}}}\) denote the cotangent bundle on \({{{\overline{X}}}}\). For \(k\gg 1\), we derive an estimate for \(\mu _{{ {\overline{X}}}}^{\textrm{Ber},{{k}}}\), the Bergman metric associated to the line bundle \({{\mathcal {L}}}^{k}:=\Omega _{{{{\overline{X}}}}}^{\otimes {{k}}}\otimes {{\mathcal {O}}}_{{{{\overline{X}}}}}((k-1)i\infty )\). For a given \(d\ge 1\), the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by \(\mu _{\textrm{Sym}^{{d}}({{\overline{X}}})}^{\textrm{FS},k}\), defines a Kähler metric on \(\textrm{Sym}^{{d}}({{\overline{X}}})\), the d-fold symmetric product of \({{{\overline{X}}}}\). Using our estimates of \(\mu _{{ {\overline{X}}}}^{\textrm{Ber},{{k}}}\), as an application, we derive an estimate for \(\mu _{\textrm{Sym}^{{d}}({{\overline{X}}}),\textrm{vol}}^{\textrm{FS},k}\), the volume form associated to the (1,1)-form \(\mu _{\textrm{Sym}^{{d}}({{\overline{X}}})}^{\textrm{FS},k}\).