QUILLEN METRICS AND SINGULAR FIBERS IN ARBITRARY RELATIVE DIMENSION

Let pi : X --> S be a holomorphic map of compact complex manifolds, wh ich is a submersion on the complement of a smooth submanifold Sigma of codimension 2, where pi degenerates quadratically. Let xi be a holomo rphic vector bundle on X. Set lambda(jxi) = (det R pi*xi)(-1). Let g( TX) be a Kahler metric on TX, let g(xi) be a Hermitian metric on xi. I f Delta = pi(Sigma), let parallel to parallel to(lambda(jxi)) be the smooth Quillen metric on Delta(jxi)\(S\Delta) associated to g(TX), g( xi). The purpose of this paper is to describe the behaviour of paralle l to parallel to(lambda(jxi)/S/Delta) near Delta. After extracting a logarithmic divergence, we describe the limit metric in terms of the Q uillen metric on the normalization of the singular fibres. To establis h our results, we make an essential use of the immersion theorem of Bi smut-Lebeau for Quillen metrics.