Faltings' construction of the K-Z connection
Let \(X\) be a complex projective, irreducible, smooth curve of genus \(g\geq 2\), and let \(L\) be a line bundle on \(X\). Write \(SU_X:=SU_X(r,L)\) for the moduli space of semi-stable vector bundles \(F\to X\) such that \(\text{rank}(F)=r\) and \(\det(F)=L\). Let \(\Theta_X\) be the ample generator of \(\text{Pic}(SU_X)\simeq{\mathbb Z}\). One may also consider the relative case \(\mathcal X\to T\) of a family of curves (of genus \(g\)) over the base \(T\), with \(\mathcal X\) smooth projective over \(T\). Fixing a line bundle \(\mathcal L\) on \(\mathcal X\), one obtains the family \(\Pi:\mathcal{SU}_{\mathcal X}\to T\) of relative moduli spaces. For a line bundle \(\Theta_{\mathcal X}\) on \(\mathcal{SU}_{\mathcal X}\) that restricts to \(\Theta_X\) for each fibre \(X\) of \(\Pi\) one may define the direct image sheaves \(\mathcal V_k:=\Pi_*\Theta_{\mathcal X}^k\), \(k\in{\mathbb Z}\). Then the \(\mathcal V_k\) are in fact vector bundles. The following theorem holds: \(\mathcal V_k\) carries a natural connection \(\nabla\) with vanishing curvature. This result was obtained by several people [\textit{A. Tsuchiya, K.Ueno} and \textit{Y. Yamada} in: Integrable systems in quantum field theory and statistical mechanics, Proc. Symp., Kyoto and Kyuzeso 1988, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010); \textit{N. J. Hitchin}, Commun. Math. Phys. 131, No. 2, 347-380 (1990; Zbl 0718.53021); \textit{S. Axelrod, S. Della Pietra} and \textit{E. Witten}, J. Differ. Geom. 33, No. 3, 787-902 (1991; Zbl 0697.53061); \textit{G. Faltings}, J. Algebr. Geom. 2, No. 3, 507-568 (1993; Zbl 0790.14019)]. In the underlying paper an attempt is made to describe the paper by Faltings. The set-up is complex analytic. One considers the moduli space \(\mathcal R\) of irreducible representations of the fundamental group of the Riemann surface \(\Sigma\) with (almost) complex structure \(J\) underlying \(X\), into \(SL(2,{\mathbb C})\). Then \(\mathcal R\) is independent of \(J\) and carries a holomorphic symplectic form \(\Omega\), and one shows that \(\mathcal R\) admits a holomorphic line bundle \(\mathcal L\) with holomorphic connection \(\nabla\) with corresponding curvature \(\Omega\). For \(k\in{\mathbb Z}\) one writes \(\nabla^k\) for the connection on \(\mathcal L^k\) induced by \(\nabla\). Let \(\tau_0\) denote the holomorphic volume form corresponding to \(\Omega\), and write \(\tau=e^f\tau_0\) for another volume form, where \(f\) is to be determined properly. There is an open set \(U\subset\mathcal R\) and a map \(\phi:U\to SU^s_X\), where \(SU^s_X\subset SU_X\) is the open set parametrising stable bundles. Reversing the complex structure \(J\mapsto -J\) one writes \(\widetilde{X}\), \(\widetilde{\phi}\), \(\ldots\), etc.\ for the objects, maps, \(\ldots\), etc.\ corresponding to \(-J\). Then the fibres of \(\phi\) and \(\widetilde{\phi}\) are transversal. Let \(\kappa_1\) denote the pull-back to \(U\) of the canonical bundle \(\kappa=\kappa_{SU_X^s}\) of \(SU_X^s\). Then \(\kappa=\Theta^{-c}\) with \(c=4\). Now, on the one hand, one has an isomorphism \(\kappa_1\overset\sim\rightarrow\mathcal L^{-c}\) and an induced connection \(\nabla^{-c}\) . On the other hand, for suitable \(f\) (thus \(\tau)\), one constructs a connection \({}^{\tau}\nabla\) on \(\kappa_1\) with suitable properties with respect to (the fibres of) \(\phi\) and \(\widetilde{\phi}\). The main point is now that one may choose \(\tau\) such that \(\nabla^{-c}={}^{\tau}\nabla\).
fundamental group of the Riemann surface
scientific article; zbMATH DE number 1203799
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