  Let M be a connection on a complex affine space with poles along a
hyperplane. In this paper, we prove that the nearby cycles of Abbes and Saito's
construction applied to M satisfy a linearity condition analogous to that
obtained by Abbes and Saito in the l-adic setting. This generalizes the fact
that in dimension one, the module produced by Abbes and Saito's construction is
a direct sum of exponential modules associated to linear forms.
