Moduli spaces of stable bundles on non-Kählerian elliptic fibre bundles over curves
The first part of the paper is a survey of known results of principal elliptic fibre bundles \(X\to B\) over a compact Riemann surface. If the fibre is the elliptic curve \(E\) one can chose a lattice \(\Gamma=\mathbb{Z} \oplus\mathbb{Z} \tau\) such that \(E=\mathbb{C}/\Gamma\) and the characteristic class of the fibration arising from the exact sequence \[ 0\to\Gamma_B \to{\mathcal O}_B\to {\mathcal O}_B(E)\to 0 \] \(({\mathcal O}_B(E)\) sheaf of germs of holomorphic maps from \(B\) to \(E)\) has the form \((d,0)\in H^2(B,\Gamma) \simeq\Gamma\), with \(d>0\). The topology of \(X\) is determined by the pair \((g,d)\), \(g\) the genus of \(B\), and \(X\) admits a Kähler metric if and only if \(d=0\) (since the first Betti number is \(2g+1\), if \(d>0)\). Then the Picard group of \(X\) is studied. The second cohomology has torsion, namely the image of \(H^2(B,\mathbb{Z}) =\mathbb{Z}\) in \(H^2(X, \mathbb{Z})\), and it has order \(d\). The group of line bundles with \(dc_1(L)=0\) is denoted by \(\text{Pic}^T(X)\), and it is shown that as an analytic manifold it is isomorphic to the disjoint union of \(\text{Pic}^i(B) \times\mathbb{C}^*\). Then it is shown that for the choice of a Gauduchon metric on \(X\) the degree map is given on \(\text{Pic}^i (B)\times \mathbb{C}^*\) by \(i-d{\log|z|\over \log|q |}\) (where \(q=e^{2\pi i\tau})\) up to a constant positive factor (independent on \(i)\). In the second part the so-called graph method for studying 2-bundles on elliptic fibred surfaces is explained in the situation of elliptic principal fibre bundles and for \(Sl(2,\mathbb{C})\)-bundles. This method associates a divisor \(G(E)\) on \(B\times\mathbb{P}^1\) (where \(B\times\mathbb{P}^1\) has to be considered as \(\text{Pic}^0_{X/B}/ \{\pm 1\}= B\times E/(\pm 1))\) which depends algebraically on the \(Sl(2,\mathbb{C})\)-bundles \(E\) and determines the behavior of the restriction of \(E\) onto the fibre over \(b\in B\). Due to the fact that \(X\) is non-Kählerian \((d>0)\), it contains generically no line bundle of non-zero degree, so it is an extension of a line bundle \(L\) of degree 0 by \(L^{-1}\), and \(\{L,L^{-1}\}\) determines a point on \(\mathbb{P}^1\). The graph is an extension of this correspondence between \(B\) and \(\mathbb{P}^1\). In the third part this is applied to a description of the moduli space of stable \(Sl(2,\mathbb{C})\)-bundles with second Chern class 1, over surfaces which are principal elliptic fibre bundles over a compact Riemann surface \(B\) of genus \(>0\). The graph is in this case (because of \(g(B)\geq 1\), \(c_2(E)=1)\) the sum of a vertical and a horizontal fibre of \(B\times\mathbb{P}^1\). The moduli space has the form \({\mathcal M}=X\times_B{\mathcal N}\), where \({\mathcal N}\to B\) is a locally trivial fibration. If \(d\) is even, \({\mathcal N}\) is isomorphic to the disjoint union of \({d\over 2}\) copies of \(B\times R\times\text{Pic}^0(B)\), where \(R\) is the annulus \(\{z\mid {1\over 2}<|z|<2\}\). If \(d\) is odd, \({\mathcal N}\) is a disjoint union of \({d-1\over 2}\) copies of this space and it has an additional component wich is a non-trivial fibration over \(B\), and the fibre is described. Finally, in the fourth part an alternative proof of a result of \textit{P. J. Braam} and \textit{J. Hurtubise} [J. Reine Angew. Math. 400, 146-172 (1989; Zbl 0669.32012)] is given. It describes the moduli of stable \(Sl(2,\mathbb{C})\)-bundles with \(c_2=1\) over primary elliptic Hopf surfaces.
scientific article; zbMATH DE number 1192193