Theta functions on moduli spaces of vector bundles
The paper is a survey about generalized theta functions as sections of ample line bundles on moduli spaces [see also \textit{G. Faltings}, J. Algebr. Geom. 18, No. 2, 309--369 (2009; Zbl 1161.14025)]. It has seven sections: \textit{1. Introduction}. Here the frame of the paper is introduced: one considers moduli of \(G\)-bundles over a curve, where \(G\) is a semisimple simply connected algebraic group and the main results are described: the Picard group of \({\mathcal M}_G\) is infinite cyclic, (the sections of these vector bundles are generalized theta functions) and the dimension of the vector space of sections is given (via Verlinde formula). \textit{2. Moduli spaces.} Here the construction of the stack \(\mathcal{M}_{r,d}\) of vector bundles of rank \(r\) and degree \(d\) over a curve is sketched. Then one describes in few words the changes if \(GL_r\) is replaced with a group as above. \textit{3. The double quotient.} Here the following setting is considered: take \(C\) a smooth projective curve over an algebraic closed field \(k\), \(x\in C\) a point and \(C^0=C\setminus \{x\}\). Then the moduli space is a double quotient: \[ \mathcal{M}_G(k)=G(C^0)\setminus G(k((t))) /G(kt) . \] One studies first the right quotient, which turns out to be an affine Grassmanian \({\mathbb D}_G\). Using \(\text{Pic}({\mathbb D}_G)\) one reduces the study of \(\text{Pic}({\mathcal M}_G)\) to \(G(C^0)\)-equivariant line bundles on \({\mathbb D}_G\). \textit{4. Line bundles on \({\mathcal M}_G\).} Two important examples corresponding to \(G=SL_r\) and \(G=Spin(r)\) (spingroup of \(SO(r)\)) are sketched. \textit{5. Construction of a line bundle} of invariant \(1\). \textit{6. The Verlinde formula.} Here the use of Verlinde formula to compute, in characteristic zero, \(\text{dim }\Gamma ({\mathcal M}_G,{\mathcal L}_c)\) for a line bundle of invariant \(c\) is explained in a simplified way. Some comments about ``das große offene Problem'' of describing geometrically these sections are given. \textit{7. The Hitchin fibration.} Here the use of the Hitchin fibration to obtain results in positive characteristic is described, in a simplified version.
scientific article; zbMATH DE number 5545551