Generalized SU(2) theta functions
In 1965, M. S. Narasimhan and C. S. Seshadri showed that the moduli space of stable rank-2 vector bundles with trivial determinant over a compact Riemann surface of genus \(g\) is diffeomorphic to the smooth real manifold parametrizing the irreducible SU(2)-representations of the fundamental group of the Riemann surface modulo conjugation. This result was generalized by V. Mehta and C. S. Seshadri in 1980, who proved that the manifold \(\Hom^{\text{irr}} (\Gamma, \text{SU(2))/SU(2)}\), associated with a Fuchsian group having only parabolic fixed points in the Siegel upper-half plane, can be regarded as a diffeomorphic model of the moduli space of certain special stable vector bundles of rank 2 with trivial determinant [cf. \textit{V. B. Mehta} and \textit{C. S. Seshadri}, Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)]. The occuring vector bundles are called parabolic bundles, and their dependence on \(\Gamma\) is expressed by specifying data \((D,k)\), where \(D\) is an effective divisor of even degree on the curve and \(k\) a certain integer. The moduli space \(M(X,D,k)\) of semistable parabolic bundles carries a natural polarizing line bundle \({\mathcal L}\) which is called the generalized theta bundle on \((M,X,D,k)\). The space \(H^ 0 (M(X,D,k), {\mathcal L})\) of global sections of \({\mathcal L}\) is commonly called the space of generalized SU(2)-theta functions (with respect to the data \((D,k))\). The aim of the present paper is to compute the dimension of the \(\mathbb{C}\)-vector space \(H^ 0 (M(X, D, k), {\mathcal L})\) in terms of \(D,k\), and the genus \(g\) of \(X\). The result is the verification of Verlinde's formula, which has recently been established for a more general situation by \textit{G. Faltings} [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]. The author's approach, in this special case, is conceptually much simpler and yields, as a byproduct, some new information about certain sheaves on projective spaces. Moreover, the author shows that the crucial trick, which M. Thaddeus invented to prove Verlinde's formula in the even more special case of \(D = \emptyset\) [cf. \textit{M. Thaddeus}, ``Stable pairs, linear systems and the Verlinde formula'', Ph. D. Thesis (Univ. Oxford 1992); see also Invent. Math. 117, No. 2, 317-353 (1994)], also works in the more general case under consideration. In the first part of the paper, the construction of the relevant moduli spaces (of parabolic bundles) and their polarizing theta bundles is given in terms of geometric invariant theory. Then the degeneration behavior of SU(2)-theta functions for families of curves is analyzed, and the result (Verlinde's formula) is derived from this analysis by applying the fact that Verlinde's formula has already been verified for the case of \(D\) being one point. The latter statement had been obtained by the author and \textit{A. Szenes} [Topology 32, No. 3, 599-609 (1993; Zbl 0798.14004)] shortly before.
scientific article; zbMATH DE number 549658
scientific article