Motivic infinite cyclic covers

Abstract

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (under certain finiteness conditions) an element in the Grothendieck ring K 0 (Varμ C ), which we call motivic infinite cyclic cover, and show its birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.2010 Mathematics Subject Classification. 32S55, 14J17, 14J70, 14H30, 32S45. Key words and phrases. infinite cyclic cover of finite type, motivic infinite cyclic cover, Milnor fiber, motivic Milnor fiber, Betti realization. 1 ∅ = I ⊆ J) of E, the fiber of the latter fibration being homeomorphic to (C * ) |I| , where |I| denotes the number of elements in the set I.Note that the punctured neighborhood T * X,E is homotopy equivalent to the boundary of the regular neighborhood T X,E , which sometimes is called the link of E in X.Definition 2.1. Infinite cyclic cover of finite type. Let ∆ : π 1 (T * X,E ) ։ Z be an epimorphism 1 , and denote by T * X,E,∆ the corresponding infinite cyclic cover (with Galois group Z) of the punctured neighborhood T * X,E of a simple normal crossing divisor E ⊂ X. For any i ∈ J, let δ i be the boundary of a small (oriented) disk transversal to the irreducible component E i . We call the infinite cyclic cover T * X,E,∆ of finite type if m i = ∆(δ i ) = 0 for all i ∈ J (see Proposition 2.4 below for a justification of the terminology).Remark 2.2. The surjectivity of the restriction of ∆ on the kernel of π 1 (T * E ) → π 1 (T E ) = π 1 (E) is equivalent to the condition gcd(m i |i ∈ J) = 1. Sometimes we omit ∆ and X in the notation and write simply T * E . The map ∆ will also be referred to as the holonomy of this infinite cyclic cover. Note also that ∆ factors through H 1 (T * E ) , so the infinite cyclic 1 The surjectivity assumption is made here solely for convenience (in which case the corresponding infinite cyclic cover is connected), all results in this paper being valid for arbitrary homomorphisms to Z. The only instance when non-surjective homomorphisms are considered is in Section 5.

Complements to ample divisors and Singularities

“…The condition that meridians are taken by to non-zero element of implies that the covering space ( ( 1 ) \ ∩ ( 1)) is 1-finite (cf. [99]) and surjectivity of the maps of the kernels implies that so is ( \ ) . Since the map in the top row in ( 14) is surjective,…”

Newton transformations and motivic invariants at infinity of plane curves

“…Using constructions of Denef-Loeser [12,13] and Guibert-Loeser-Merle [19], a motivic Milnor fiber of f at a point x ∈ I for a value c is introduced in [31, §4] (see also [16] and [28]). We give in Theorem 3.9 its computation in dimension 2 in a full generality.…”

“…Working over a characteristic zero field k, with P and Q polynomials with coefficients in k, using the motivic integration theory, introduced by Kontsevich in [23], and more precisely constructions of Denef-Loeser in [11,12,14] and Guibert-Loeser-Merle in [19], the second author in [31] and Nguyen-Takeuchi in [28] (see also [16] and [39]) defined a motivic Milnor fiber of f at x and a value c, denoted by S f,x,c (section 3.3). It is an element of M Gm Gm , a modified Grothendieck ring of varieties over k endowed with an action of the multiplicative group G m of k. When k is the field of complex numbers, it follows from Denef-Loeser results that the motive S f,x,c is a "motivic" incarnation of the topological Milnor fiber F x,c endowed with its monodromy action T x,c .…”