Intersection numbers on moduli spaces and symmetries of a Verlinde formula. II.
type: Item
instance of: scholarly article
title: Intersection numbers on moduli spaces and symmetries of a Verlinde formula. II.
zbMATH Open document ID: 1044.14012
DOI: 10.1016/S0393-0440(00)00021-8
author: R. Smith
published in: Journal of Geometry and Physics
publication date: 2004-01-07T00:00:00Z
review text: For part I of this paper see \textit{R. Herrera} and \textit{S. Salaman} [Commun. Math. Phys. 188, No.~3, 521--534 (1997; Zbl 0886.14014)]. The author studies a certain moduli space \(M_{g,n}\) of orthogonal vector bundles of rank \(2n\) over a hyperelliptic Riemann surface \(\Sigma\) of genus \(g\). The space \(M_{g,n}\) is one of those introduced by \textit{S. Ramanan} [Proc. Indian Acad. Sci. Math. Sci. 90, No. 2, 151--161 (1981; Zbl 0541.04012)] and was identified by him with the variety of \((g+1-n)\)-dimensional subspaces of \({\mathbb C}^{2g+2}\) which are isotropic with respect to a pencil of quadratic forms canonically associated with \(\Sigma\) [op. cit., theorem 3]. In particular \(M_{g,1}\) is the Jacobian of \(\Sigma\) and \(M_{g,2}\) is the moduli space of stable holomorphic vector bundles over \(\Sigma\) of rank \(2\) and fixed determinant of odd degree. The author uses Ramanan's description to obtain geometric and topological information about \(M_{g,n}\). In fact \(M_{g,n}\) is a projective variety and admits a positive line bundle \(L\). It is natural to study the numbers \(\dim H^0(M_{g,n},{\mathcal O}(L^k))\) as generalisations of the Verlinde numbers, and also the intersection theory of \(M_{g,n}\), using the methods of part I of this paper. As a result of this study, the author obtains vanishing and symmetries for the holomorphic Euler characteristics of certain virtual vector bundles, allowing a computation of the generalised Verlinde numbers. In the case \(n=g-1\), he obtains also some formulae for intersection numbers and hence the vanishing of the Pontrjagin numbers, the \(\hat A\)-genus and the top two Chern classes of \(M_{g,g-1}\). This generalises well known results for \(M_{g,2}\).
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 1542452
zbMATH Keywords: hyperelliptic Riemann surface
zbMATH Keywords: orthogonal vector bundle
zbMATH Keywords: moduli space
zbMATH Keywords: Verlinde formula
reviewed by: P. E. Newstead
MaRDI profile type: Publication
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full work available at URL: s0393-0440(00)00021-8
OpenAlex ID: W1968229583
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description: scientific article; zbMATH DE number 1542452