A Local Families Index Formula for $${\overline{\partial}}$$ -Operators on Punctured Riemann Surfaces

Abstract

Abstract. Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of ∂-operators on the Teichmüller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space Mg,n in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.

Families Index for Manifolds with Hyperbolic Cusp Singularities

“…In a coordinate chart near the boundary -with y i , z j denoting coordinates along the base Y of the fibration and the fibre Z respectively -the set of vector fields of finite length with respect to a φ-hc metric (denoted V φ-hc ) is locally spanned by (2) x∂ x , ∂ y i , 1 x ∂ z j .…”

“…(All of these assumptions hold for a natural family of ∂ operators on Riemann surfaces with cusps as will be explained in a companion paper [2].) Then, in this case, we have (4.12)…”

“…This curvature corresponds to the two form part of the family index. In [2], we use our results to get a formula for the 'full' family index, generalizing in this way the formula of Takhtajan and Zograf.…”

Some Index Formulae on the Moduli Space of Stable Parabolic Vector Bundles

“…However, on punctured Riemann surfaces, the geometry at infinity is sufficiently simple to allow explicit computations. In [2], using the generalization of Vaillant's index theorem to families from [1], this fact was put to use to get a local index theorem in terms of the Mumford-Morita-Miller classes for families of ∂-operators parametrized by the moduli space of Riemann surfaces of genus g with n marked points. Using heat kernel techniques as in [9] (see also [6, Sections 9 and 10]), it was also possible to give an alternative proof of the formula of Takhtajan and Zograf [37] for the curvature of the Quillen connection defined on the corresponding determinant line bundle.…”

“…This leads to local families index formulae for the families ∂ E and ∂ E (Theorems 5.2 and 5.4). In both cases, we are also able, as in [2], to define the Quillen metric on the determinant line bundle via heat kernel techniques and identify its curvature with the 2-form part of the index formula (Theorems 6.1 and 6.4). For the determinant line of the family ∂ E , our formula agrees with that of Takhtajan and Zograf [38,Theorem 2].…”