On arbitrary spacetimes, we study the characteristic Cauchy problem for Dirac
fields on a light-cone. We prove the existence and uniqueness of solutions in
the future of the light-cone inside a geodesically convex neighbourhood of the
vertex. This is done for data in L2 and we give an explicit definition of
the space of data on the light-cone producing a solution in H1. The method
is based on energy estimates following L. H\"ormander (J.F.A. 1990). The data
for the characteristic Cauchy problem are only a half of the field, the other
half is recovered from the characteristic data by integration of the
constraints, consisting of the restriction of the Dirac equation to the cone. A
precise analysis of the dynamics of light rays near the vertex of the cone is
done in order to understand the integrability of the constraints; for this, the
Geroch-Held-Penrose formalism is used.Comment: 39 pages. An error in lemma 3.1 in the first version has been
corrected. Moreover, the treatment of the constraints (restriction of the
equations to the null cone) has been considerably extended and is now given
in full details. To appear in Journal of Hyperbolic Differential Equation
