On the poles of zeta functions for finite morphisms between normal surfaces

Technical Summary: On the Poles of Zeta Functions for Finite Morphisms Between Normal Surfaces

Problem Statement
Motivic and topological zeta functions are fundamental invariants in singularity theory, originally defined for functions on smooth varieties and later extended to pairs $(D, W)$ consisting of a divisor $D$ (resembling a function) and a divisor $W$ (representing a differential form) on a possibly singular variety. A central question in this field is understanding the relationship between the sets of poles of these zeta functions under finite morphisms. Specifically, given a finite morphism $\rho: (S, o) \to (\bar{S}, \bar{o})$ between normal surfaces, how do the poles of the zeta functions associated with divisors on the target $\bar{S}$ relate to those on the source $S$? While an equality of the form $Z_{\text{top}}(\bar{S}) = |\text{deg}(\rho)| \cdot Z_{\text{top}}(S)$ holds for specific quotient maps with trivial differential forms, the general behavior under finite morphisms, particularly regarding the inclusion of pole sets, remains unclear.

Methodology
The authors employ a combination of geometric resolution techniques and arithmetic analysis of numerical data:

1. Embedded Q-Resolutions: Instead of standard embedded resolutions, the paper utilizes embedded Q-resolutions, which are partial toric resolutions arising from weighted blow-ups. These allow the ambient space to have cyclic quotient singularities, which is natural when dealing with quotient surfaces.
2. Commutative Diagrams and Jung's Construction: To relate the source and target, the authors construct a commutative diagram involving the finite morphism $\rho$, a resolution $\pi$ of the target, and an induced Q-resolution $\tilde{\pi}$ of the source. This construction is inspired by Jung's method, utilizing fiber products and normalization to ensure the source resolution maps finitely to the target resolution.
3. Numerical Data and $\alpha$-Values: The analysis relies heavily on the numerical data $(N_i, \nu_i)$ of exceptional components (multiplicities and log discrepancies) and the derived $\alpha$-values. These values, defined as $\alpha_P = \nu_j - (\nu_i/N_i)N_j$ (normalized by the order of the singularity), determine the residues of candidate poles.
4. Residue Analysis: The authors analyze the conditions under which candidate poles (values $s_0 = -\nu_i/N_i$) become actual poles. This involves studying the "rupture components" (exceptional curves intersecting at least three other components with specific $\alpha$-value properties) and the behavior of residues in the Grothendieck ring of varieties.

Key Contributions and Results

• General Inclusion for Motivic Zeta Functions (Theorem 1):
  The paper establishes a general inclusion result for motivic zeta functions. If $\rho: S \to \bar{S}$ is a finite morphism, and divisors $D, W$ on $\bar{S}$ are pulled back to $D = \rho^* \bar{D}$ and $W = \rho^* \bar{W} + \text{Ram}_\rho$ on $S$, then the set of poles of the motivic zeta function on the target is contained in the set of poles on the source:
  $$P_{\text{mot}}^{\bar{S}}(\bar{D}, \bar{W}) \subseteq P_{\text{mot}}^{S}(D, W).$$
  Furthermore, the order of a pole of order two is preserved. The authors provide examples (e.g., Example 3.2) showing this inclusion is strict in general and that the reverse inclusion does not hold.

• Non-Inclusion for Topological Zeta Functions:
  Unlike the motivic case, the paper demonstrates that no general inclusion holds for topological zeta functions. Examples (4.1 and 3.2) show that a pole may exist in the source but not the target, or vice versa. This is attributed to the fact that the condition for a component to be a "rupture component" (which contributes to poles) is not preserved under pushforward in the topological setting.

• Equality for Quotient Maps with Trivial Differential Forms (Theorem 2):
  A stronger result is proven for the specific case where the morphism is a quotient map $\rho: \mathbb{C}^2 \to \mathbb{C}^2/G$ induced by a finite abelian group $G$, and the differential form divisor on the source is trivial ($W=0$). In this setting, the authors construct a specific embedded Q-resolution that satisfies the $\alpha$-condition (where $\alpha$-values are bounded by 1). Under these conditions, the sets of poles for both motivic and topological zeta functions coincide, and the orders of poles are identical:
  $$P_{\text{mot}}^{\mathbb{C}^2/G}(\bar{D}, -B_\rho) = P_{\text{top}}^{\mathbb{C}^2/G}(\bar{D}, -B_\rho) = P_{\text{top}}^{\mathbb{C}^2}(D, 0) = P_{\text{mot}}^{\mathbb{C}^2}(D, 0).$$

• Multiplicative Relation for Invariant Divisors (Theorem 3):
  The paper recovers the multiplicative relation $Z_{\text{top}}(\bar{S}) = |G| \cdot Z_{\text{top}}(S)$ (Equation 1) under the condition that every analytically irreducible component of both $D$ and $W$ is $G$-invariant. The authors show that if components are permuted by the group action (a pathological case where normal crossing is not preserved in the quotient), this equality fails.

• Characterization of Poles (Theorem 1.11 and 7.1):
  The authors provide a geometric characterization of the poles of motivic zeta functions for arbitrary $W$ on normal surfaces using embedded Q-resolutions. A value $s_0$ is a pole if and only if it corresponds to:

  1. A strict transform component in $\bar{D} \cap \bar{W}$ or $\bar{D} \setminus \bar{W}$ (with $\nu_i=1$);
  2. A non-rational exceptional curve;
  3. A cycle of rational exceptional curves; or
  4. A rational exceptional curve containing at least three points with $\alpha$-values $\neq 1$ (generalizing the concept of rupture components).

Significance
The paper clarifies the complex relationship between zeta functions of source and target under finite morphisms, distinguishing sharply between the motivic and topological levels. It demonstrates that while motivic poles exhibit a containment property, topological poles do not, highlighting the sensitivity of topological invariants to the specific geometry of the resolution. The work extends the theory of $\alpha$-values and rupture components to the setting of Q-resolutions and singular ambient surfaces, providing a robust framework for analyzing zeta functions on quotient singularities. The results refine the understanding of when simple multiplicative relations (like Equation 1) hold, identifying the precise geometric obstructions (such as the permutation of curve components by the group action) that prevent such equalities. Additionally, the paper connects these findings to log canonical models, suggesting that the poles of zeta functions correspond to the components of the log canonical model of the pair.