Novikov type inequalities for differential forms with non-isolated zeros

TL;DR

This paper extends Novikov inequalities to include non-isolated critical points and incorporates flat bundle twisting, providing new analytic proofs of Bott's degenerate Morse inequalities using spectral estimates.

Contribution

It generalizes Novikov inequalities to non-isolated critical points and introduces twisting by flat bundles, with a new analytic proof approach.

Findings

Extended Novikov inequalities to non-isolated critical points

Incorporated flat bundle twisting into inequalities

Provided an explicit spectral estimate for the Laplacian

Abstract

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.