Bergman kernel on Riemann surfaces and Kaehler metric on symmetric products

Abstract

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\ge 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle ${\Omega }_X^{\otimes k}$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding Bergman metric on $X$ in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of $X$ into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of ${\Omega }_X^{\otimes k}$. The Fubini-Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of $X$. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of ${\Omega }_X^{\otimes k}$ and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on $X$.

Final version; Int. Jour. Math. (to appear)

References in corpus (2)

• One-vortex moduli space and Ricci flow
• Moduli of vortices and Grassmann manifolds