Formulas for monodromy

Abstract

Given a family X of complex varieties degenerating over a punctured disk, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of X . Our first main result is that the motivic nearby fiber of X can be computed by first stratifying X into locally closed subvarieties that are nondegenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when X is a family of nondegenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.

On the Mixed Hodge Structures of the Intersection Cohomology Stalks of Complex Hypersurfaces

“…This result follows from a deep argument combining the computation of the motivic nearby fiber and some combinatorial results (see Stapledon [19], Saito-Takeuchi [16], Guibert-Loeser-Merle [6]). This proposition implies that for an eigenvalue λ = 1, the filtration…”

“…In this paper, we reveal a new relationship between the mixed Hodge structures of the stalks of the intersection cohomology complexes (we call them IC stalks for short) and the Milnor monodromies, by using the results on motivic Milnor fibers shown by Matsui-Takeuchi [11] and on motivic nearby fibers by Stapledon [19]. For a natural number n ≥ 2, let f ∈ C[x 1 , .…”

“…and dim gr W r H n−2 (L; Q) = dim gr W 2(n−1)−r H n−1 (L; Q) = J 1 n−r−1 . We prove Theorem 1.1 by combining the results of Stapledon [19] with that of Matsui-Takeuchi [11]. For this purpose, we deform the hypersurface V to a suitable one which can be compactified in P n nicely.…”

Triangulations of simplices with vanishing local h-polynomial

“…Recalling that all vertices of C have excess d − 2, we note that π 1 (v 2 ) > 0 and π 2 (v 1 ) > 0, otherwise they would be contained in F 1 ∩ F 2 . Therefore, by (8),…”

“…Our investigation into the structure of triangulations with vanishing local hpolynomial is motivated by recent connections to algebraic and arithmetic geometry. Local h-polynomials appear prominently in formulas for dimensions of homology groups of intersection complexes for toric morphisms [3] and multiplicities of eigenvalues of monodromy [6,8]. In Igusa's p-adic monodromy conjecture [5], and the motivic generalization of Denef and Loeser [4], the essential question is understanding whether or not these multiplicities vanish.…”