Witten Deformation and Polynomial Differential Forms

摘要

As is well-known, the Witten deformation d (sub h) of the De Rham complex211u001ecomputes the De Rham cohomology. In this paper, the authors study the Witten 211u001edeformation on a noncompact manifold M and restrict it to differential forms 211u001ewhich behave polynomially near infinity. Such polynomial differential forms 211u001enaturally appear on manifolds with a cylindrical structure. The authors prove 211u001ethat the cohomology of the Witten deformation d(sub h) acting on the complex of 211u001ethe polynomially growing forms (depends on h and) can be computed as the relative 211u001ecohomology of the apir (M,F) where F is a negative remote fiber of h. The authors 211u001eshow that the assumptions of their main theorem are satisfied in a number of 211u001einteresting special cases, including generic real polynomials on R(sup n).