On Maximal order poles of generalized topological zeta functions

Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties

“…Note that for all a ≥ 1, the candidate poles − a+2 m and − ν 1 m 1 induce eigenvalues of the characteristic polynomial of the monodromy given in (26). In this case, D 2 = (a − 1)L always provides an allowed differential 3-form in the sense of [35], see also the recent work [3].…”

“…In this case, the relative canonical divisor is K π = 4E. Let us fix coordinates (x, y) on X around the origin and coordinates (u, v) on Y around Q, such that π is given by the substitution x = uv 2 and y = v 3 . More details about weighted blow-ups are given e.g.…”

“…On the other hand, ord t K π (ψ) = 4k 2 3 , which is much easier to calculate. In order to verify (12), note that ord t ω [3] X is the zero function and that ord t ω [3] Y (ψ) = −4 min{k 1 , k 2 }, using the calculation above and the fact that (du ∧ dv) ⊗3 generates ω…”

“…Strictly speaking one has to choose a d 2 /βth root of unity to calculate S G 2 (N 2 , ν 2 ) in Theorem 3. Since all the terms appearing in M i,j above are dth roots of unity, one gets 3(Ns + ν) 3 .…”