Moduli spaces of stable bundles on non-Kählerian elliptic fibre bundles over curves
type: Item
instance of: scholarly article
title: Moduli spaces of stable bundles on non-Kählerian elliptic fibre bundles over curves
zbMATH Open document ID: 0933.14022
author: Andrei Teleman
published in: Expositiones Mathematicae
publication date: 2000-04-04T00:00:00Z
review text: 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 \((d>0)\),…
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 1192193
zbMATH Keywords: moduli of stable bundles
zbMATH Keywords: elliptic fibre bundles
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: cohomology
zbMATH Keywords: graph method
zbMATH Keywords: elliptic Hopf surfaces
reviewed by: Herbert Kurke
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1192193