  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 $L^2$ and we give an explicit definition of
the space of data on the light-cone producing a solution in $H^1$. 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.
