  We generalize the Novikov inequalities for 1-forms in two different
directions: first, we allow non-isolated critical points (assuming that they
are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the
inequalities by means of twisting by an arbitrary flat bundle.
  The proof uses Bismut's modification of the Witten deformation of the de Rham
complex; it is based on an explicit estimate on the lower part of the spectrum
of the corresponding Laplacian.
  In particular, we obtain a new analytic proof of the degenerate Morse
inequalities of Bott.
