Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci

Abstract

We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part the question concerning the motivic nearby cycles of restriction functions in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.

Keywords

• arc spaces
• contact loci
• motivic zeta function
• motivic Milnor fiber
• motivic nearby cycles
• Newton polyhedron
• nondegeneracy
• sheaf cohomology with compact support