Witten Deformation and Polynomial Differential Forms

As is well known, the Witten deformation d_h of the De Rham complex computes the De Rham cohomology. In this paper, we study the Witten deformation on noncompact manifolds and restrict it on differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with the cylindrical structure. We prove that the cohomology of the Witten deformation d_h acting on the complex of the polynomially growing forms (depends on h and) can be computed as the cohomology of the negative remote fiber of h. We show that the assumptions of our main theorem are satisfied in a number of interesting special cases, including generic real polynomials on ℝⁿ.