On the sizes of the Jordan blocks of monodromies at infinity

Abstract

We obtain general upper bounds of the sizes and the numbers of Jordan blocks for the eigenvalues λ = 1 in the monodromies at infinity of polynomial maps.

Number of Jordan blocks of the maximal size for local monodromies

“…For λ = 1 and n = 1, Theorem 3 simply gives a well-known formula ν 1 g0,1 = r g0 − 1, where r g0 is the number of analytic local irreducible components of g −1 0 (0). Theorem 2 improves a result of Matui and Takeuchi [MT12], where the number is bounded by dim C j f,λ in the case of monodromies at infinity of polynomial maps with λ = 1 (since…”

“…This assertion also follows from a theorem in [Ste77] for the mixed Hodge numbers of the Milnor cohomology in the nondegenerate Newton boundary case with dim X = 2. (This example shows that the estimate in [MT12], which is given by dim C j f,λ , is not very good in general.) Note that some related argument using a Q-resolution is given in [Mar12].…”

“…The rank of the differential of B • f,λ , as well as the difference between dim H j B • f,λ and dim C j f,λ , can be quite large, as is seen in the case of Example 4.4 below. This example shows that, even in the nondegenerate Newton boundary case, we have to apply many blow-ups in order to get a divisor with normal crossings (in the usual sense) by taking a suitable subdivision of the dual fan, and the estimate in [MT12] may become rather bad (unless the dual fan is already smooth, i.e. consisting of simplicials generated by integral vectors with determinant 1).…”

On the monodromies and the limit mixed Hodge structures of families of algebraic varieties

“…. , x ± n ] the monodromy of Y around the origin 0 ∈ C is nothing but the monodromy at infinity of the polynomial map g : (C * ) n −→ C. In this sense, our setting is a vast generalization of the classical ones of [3], [11], [13], [20], [21], [22], [24], [27], [30], [31], [33], [39] etc. For v ∈ Z n by the Laurent expansion a v (t) = j∈Z a v,j t j (a v,j ∈ C) of the rational function a v (t) we set o(v) := ord t a v (t) = min{j | a v,j = 0}.…”