This talk gives a review on how complex geometry and a Lagrangian formulation
of 2-d conformal field theory are deeply related. In particular, how the use of
the Beltrami parametrization of complex structures on a compact Riemann surface
fits perfectly with the celebrated locality principle of field theory, the
latter requiring the use infinite dimensional spaces. It also allows a direct
application of the local index theorem for families of elliptic operators due
to J.-M. Bismut, H. Gillet and C. Soul\'{e}. The link between determinant line
bundles equipped with the Quillen\'s metric and the so-called holomorphic
factorization property will be addressed in the case of free spin j b-c
systems or more generally of free fields with values sections of a holomorphic
vector bundles over a compact Riemann surface.Comment: Actes du Colloque "Complex Geometry '98