Tempered currents and the cohomology of the remote fiber of a real polynomial map
Abstract
Let $p:R^n\to R$ be a polynomial map. Consider the complex $S'\Omega^*(\RR^n)$ of tempered currents on $R^n$ with the twisted differential $d_p=ddp$ where $d$ is the usual exterior differential and $dp$ stands for the exterior multiplication by $dp$. Let $t\in R$ and let $F_t=p^{1}(t)$. In this paper we prove that the reduced cohomology $\tilda H^k(F_t;C)$ of $F_t$ is isomorphic to $H^{k+1}(S'\Omega^*(\RR^n),d_p)$ in the case when $p$ is homogeneous and $t$ is any positive real number. We conjecture that this isomorphism holds for any polynomial $p$, for $t$ large enough (we call the $F_t$ for $t >> 0$ the remote fiber of $p$) and we prove this conjecture for polynomials that satisfy certain technical condition. The result is analogous to that of A. Dimca and M. Saito, who give a similar (algebraic) way to compute the reduced cohomology of the generic fiber of a complex polynomial.
 Publication:

arXiv eprints
 Pub Date:
 April 1997
 arXiv:
 arXiv:alggeom/9704001
 Bibcode:
 1997alg.geom..4001B
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 LaTeX 2e