Faltings' construction of the K-Z connection
type: Item
instance of: scholarly article
title: Faltings' construction of the K-Z connection
zbMATH Open document ID: 0909.14018
author: T. R. Ramadas
published in: Communications in Mathematical Physics
publication date: 1999-03-29T00:00:00Z
review text: 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 \(J\) a…
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 1203799
zbMATH Keywords: connection
zbMATH Keywords: relative moduli spaces
zbMATH Keywords: vanishing curvature
zbMATH Keywords: fundamental group of the Riemann surface
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
full work available at URL: s002200050417
OpenAlex ID: W1964205727
DOI: 10.1007/S002200050417
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description: scientific article; zbMATH DE number 1203799