The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

For a complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet of a vector bundle along the submanifold associates the L²-optimal holomorphic extension of it to the ambient manifold. When the vector bundle is given by big tensor powers of a positive line bundle, we give an asymptotic formula for this extension operator.