A local index theorem for families of $$\bar \partial $$ -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

Abstract

We prove a local index theorem for families of\(\bar \partial \)-operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmuller spaceTg,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kahler form of a new Kahler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.