  As for any symmetric space the tangent space to Siegel upper-half space is
endowed with an operation coming from the Lie bracket on the Lie algebra. We
consider the pull-back of this operation to the moduli space of curves via the
Torelli map. We characterize it in terms of the geometry of the curve, using
the Bergman kernel form associated to the curve. It is known that the second
fundamental form of the Torelli map outside the hyperelliptic locus can be seen
as the multiplication by a certain meromorphic form. Our second result says
that the Bergman kernel form is the harmonic representative - in a suitable
sense - of this meromorphic form.
