  We study the asymptotics of Ohsawa-Takegoshi extension operator and
orthogonal Bergman projector associated with high tensor powers of a positive
line bundle.
  More precisely, for a fixed complex submanifold in a complex manifold, we
consider the operator which associates to a given holomorphic section of a
positive line bundle over the submanifold the holomorphic extension of it to
the ambient manifold with the minimal $L^2$-norm. When the tensor power of the
line bundle tends to infinity, we obtain an explicit asymptotic expansion of
this operator. This is done by proving an exponential estimate for the
associated Schwartz kernel and showing that this Schwartz kernel admits a full
asymptotic expansion. We prove similar results for the projection onto
holomorphic sections orthogonal to those which vanish along the submanifold.
