An identification of the connections of Quillen and Beilinson-Schechtman
Let \(\pi:X \to S\) be a family of compact Riemann surfaces, \(E\) a holomorphic vector bundle on \(X\) and \(\lambda_ E = \text{det} (R \pi_ *E)\) the associated determinant bundle. A few years ago, \textit{A. Beilinson} and \textit{V. Schechtman} [Commun. Math. Phys. 188, No. 4, 651- 701 (1988; Zbl 0665.17010)] gave a construction for a connection \(\nabla_{BS}\) determined by any given \(C^ \infty\) connections on \(T_{X/S}\) and \(E\). They also conjectured that, when the connections on \(T_{X/S}\) and \(E\) arise from Hermitian metrics, \(\nabla_{BS}\) coincides with the connection on \(\lambda_ E\) associated to the Quillen metric on \(\lambda_ E^{-1}\). In the present paper, the authors prove this conjecture. The result is of particularly interest since the methods of construction of the two connections are quite different, that of \(\nabla_{BS}\) having a geometric and sheaf cohomological flavour, while Quillen's is highly analytical in nature.
scientific article; zbMATH DE number 529859
scientific article