Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces

Abstract

In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichm\"uller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichm\"uller space, $T_{infty}$, was shown to carry on it a natural action of the universal commensurability mapping class group, $MC_{\infty}$. In this paper we identify an interesting cofinal sub-system corresponding to the tower of finite-sheeted characteristic coverings over any fixed base surface. Utilizing a certain subgroup inside $MC_{\infty}$, (associated intimately to the inverse system of characteristic coverings), we can now descend to an inductive system of moduli spaces, and construct the direct limit ind-variety $M_{\infty}$. Invoking curvature properties of Quillen metrics on determinant bundles, and naturality under finite coverings of Weil-Petersson forms, we are able to construct on $M_{\infty}$ the natural sequence of determinant of cohomology line bundles, as well as the Mumford isomorphisms connecting these.