Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Abstract

We study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of C n by using the theory of motivic Milnor fibers. The results will be explicitly described by the mixed volumes of the faces of Newton polyhedrons.

Motivic Milnor Fibers and Jordan Normal Forms of Milnor Monodromies

“…We thus find a striking symmetry between local and global. Finally, let us mention that in [7] the results for the other eigenvalues λ = 1 in this paper were already generalized to the monodromies over complete intersection subvarieties in C n .…”

On the sizes of the Jordan blocks of monodromies at infinity

“…For example, Dimca-Saito [3] obtained an upper bound of the sizes of Jordan blocks for the eigenvalue 1 in Φ ∞ j . Recently in [12] we obtained very explicit formulas which express the Jordan normal forms of Φ ∞ j in terms of the Newton polyhedra at infinity of f (see [13] and [5] for the further developments). However they are applicable only to convenient polynomials f which are non-degenerate at infinity.…”

Invertible polynomial mappings via Newton non-degeneracy

“…The following definition of non-degeneracy is inspired from Oka's work [23] on complex local complete intersections and from the definition used by Matsui-Takeuchi [19] and Esterov-Takeuchi [11] in the global setting of complex polynomials. It was proved in [23] that, in the complex context, this is a generic condition.…”