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

Abstract

Abstract. We study natural families of ∂-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

Ricci Flow and the Determinant of the Laplacian on Non-Compact Surfaces

“…where the renormalization is carried out using x. From [1] and [38] (see [3,Appendix]) we know that, as t → 0,…”

“…Assume without loss of generality that there is only one cusp end. Recall from [3,Appendix] that the asymptotic expansion (2.4) is derived by analyzing the restriction of the integral kernel K of the heat kernel to the manifold with corners…”

“…The same 'even-ness' is true in the expansion of the heat kernel of a F-hc metric at the boundary face B tf . It follows (see [3,Appendix]) that the asymptotics at the corner B cf ∩ B tf do not contribute to a k for k even.…”

“…α 0 = R 24π ω (τ ). Because ω (τ ) − ω i (τ ) = O(x 2 i ), we then see from Lemma 2.6 as well as the construction of the heat kernel in [38] for cusps (see [3,Appendix]) and [1] for funnels that…”

“…This method has the advantage of being very flexible and systematic. It has been used by Andrew Hassell in his proof of the Cheeger-Müller theorem [18], and in recent work of two of the authors [4,5,3].…”