---
title: 'Birational Zeta Function: Invariants & Monodromy'
url: https://www.emergentmind.com/topics/birational-zeta-function
type: topic
---

# Birational Zeta Function: Invariants & Monodromy

Searching arXiv for the cited papers on birational zeta functions and related motivic zeta constructions.
Search query: `2401.17772 birational invariance motivic zeta functions K-trivial varieties`
Birational zeta function denotes a birationally invariant zeta-type construction attached to degenerations or divisors. In one precise sense, it is the motivic zeta function of a smooth, proper \(K=k((t))\)-variety with trivial canonical bundle, viewed as a birational invariant in a localized equivariant Grothendieck ring and used to analyze poles, monodromy, and obstructions to smooth fillings. In another precise sense, it is a birational analog of the Denef–Loeser motivic zeta function for a reduced divisor on a smooth complex variety, defined through dlt modifications and birational equivalence classes. A related predecessor is the motivic infinite cyclic zeta function associated with an SNC divisor together with holonomy and log discrepancy data [2401.17772] [2509.03352] [1702.06590].

## 1. Terminological range and basic frameworks

Recent work uses the term in several closely related but non-identical settings. The common feature is that a zeta-type series is constructed so that its coefficients or its rational expression are stable under an appropriate birational equivalence relation.

| Framework | Input | Coefficient ring |
|---|---|---|
| \(K\)-trivial degeneration zeta | Smooth, proper, geometrically connected \(K\)-variety with \(K_X \sim 0\) and a volume form \(\omega\) | \(\mathcal M_k^{\hat\mu}\), or the localization \(R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}\) |
| Minimal-model birational zeta | Smooth quasi-projective \(X/\mathbb C\) and a non-zero reduced divisor \(D\) | \(\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket\) |
| Motivic infinite cyclic zeta | SNC divisor \(E\) on a smooth complex quasi-projective variety, with holonomy \(\Delta\) and discrepancies \(\nu\) | \(M_{\hat\mu}\llbracket T\rrbracket_{sr}\) |

In the \(K\)-trivial setting, the series measures how \(X\) degenerates at \(t=0\), and the use of \(\hat\mu\)-equivariant coefficients records monodromy through \(\hat\mu\)-torsors. In the divisor setting, the construction is MMP-based: log resolutions are replaced by dlt modifications or minimal models, and ordinary Grothendieck classes are replaced by birational equivalence classes. In the infinite cyclic cover setting, the construction is formulated from punctured neighborhoods of SNC divisors and is birationally invariant under blow-up relations. All three frameworks are distinct from Hasse–Weil zeta functions, which count rational points over finite fields rather than encode degeneration or birational data [2401.17772] [2509.03352] [1702.06590].

## 2. Motivic zeta functions of \(K\)-trivial varieties

Let \(k\) be a field of characteristic zero, \(R=k[t]\), and \(K=k((t))\). The profinite group scheme of roots of unity is \(\hat\mu=\varprojlim \mu_n\). The relevant coefficient ring is the equivariant Grothendieck ring \(K_0^{\hat\mu}(\mathrm{Var}/k)\), localized by the Lefschetz motive \(\mathbb L=[\mathbb A^1]\), giving \(\mathcal M_k^{\hat\mu}=K_0^{\hat\mu}(\mathrm{Var}/k)[\mathbb L^{-1}]\). For technical birational statements one further localizes to
\[
R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}.
\]
If \(X/K\) is smooth, proper, geometrically connected, of pure dimension \(d\), with trivial canonical bundle \(K_X\sim 0\), and \(\omega\) is a volume form on \(X\), then the motivic zeta function is
\[
Z_{X,\omega}(T)=\sum_{n>0}\left(\int_{X\otimes_K K(n)}|\omega|\right)T^n \in \mathcal M_k^{\hat\mu}[T],
\]
where \(K(n)=k((t^{1/n}))\). If \(\lambda\in K^\times\), then
\[
Z_{X,\lambda\omega}(T)=Z_{X,\omega}(\mathbb L^{-\operatorname{ord}_t(\lambda)}T).
\]
Thus rescaling the volume form shifts the \(T\)-variable by a power of \(\mathbb L\) [2401.17772].

The rational form is obtained from an snc-model \(\mathcal X\) of \(X\), with special fiber
\[
\mathcal X_k=\sum_{i\in I}N_iE_i.
\]
For \(\emptyset\neq J\subset I\), one sets
\[
E_J=\bigcap_{j\in J}E_j,\qquad E_J^\circ=E_J\setminus \bigcup_{i\notin J}E_i,
\]
and \(N_J=\gcd\{N_j\mid j\in J\}\). If \(\bar E_J^\circ\) denotes the pullback of \(E_J^\circ\) to the normalization of \(\mathcal X\times_R R(N_J)\), then \(\bar E_J^\circ\to E_J^\circ\) is a \(\mu_{N_J}\)-torsor. Viewing \(\omega\) as a rational section of the logarithmic relative canonical bundle, one writes
\[
\operatorname{div}_{\mathcal X}(\omega)=\sum_{i\in I}\nu_iE_i.
\]
The Denef–Loeser-style formula is
\[
Z_{X,\omega}(T)=\sum_{\emptyset\neq J\subset I}[\bar E_J^\circ](\mathbb L-1)^{|J|-1}\prod_{j\in J}\frac{\mathbb L^{-\nu_j}T^{N_j}}{1-\mathbb L^{-\nu_j}T^{N_j}}.
\]
This exhibits rationality and makes the candidate poles visible through the denominators \(1-\mathbb L^{-\nu_j}T^{N_j}\) [2401.17772].

The same paper defines a generalized zeta function \(Z_{\mathcal X,\omega}(T)\) for germs \((F,\omega)\) of canonical forms on function fields \(F/K\) and proves that it is independent of the chosen snc-model. For smooth proper \(K\)-trivial \(X\), this generalized construction reduces to the original \(Z_{X,\omega}(T)\).

## 3. Birational invariance and monodromy in the \(K\)-trivial setting

The central birational invariance statement is the following. If \(X\) and \(X'\) are smooth, proper, geometrically connected \(K\)-schemes with \(K_X\sim 0\sim K_{X'}\), and \(f\colon X'\dashrightarrow X\) is birational, then for a volume form \(\omega\) on \(X\) and the uniquely induced volume form \(\omega'\) on \(X'\), one has
\[
Z_{X,\omega}(T)=Z_{X',\omega'}(T)\quad \text{in }R.
\]
The proof proceeds by constructing \(Z\) for germs \((F,\omega)\) and proving independence of snc-model via weak factorization tailored to \(\omega\)-elementary blow-ups. This bypasses the lack of a change-of-variables formula for motivic integrals of canonical forms over \(K\) [2401.17772].

The same framework organizes the monodromy problem. For a smooth proper \(Y/K\), the \(l\)-adic characteristic polynomial
\[
P_{Y,m,\sigma}(u)=\det(u-\sigma\mid H^m(Y\times_K K^a,\mathbb Q_l))
\]
is cyclotomic and independent of \(l\) and of the chosen topological generator \(\sigma\in I_K\); it is denoted \(P_{Y,m}(u)\). For an snc-model with special fiber \(\sum N_iE_i\), the A’Campo-type formula is
\[
\prod_{m\ge 0}P_{Y,m}(u)^{(-1)^{m+1}}=\prod_{i\in I}(u^{N_i}-1)^{-\chi(E_i)}.
\]
This ties multiplicities \(N_i\) directly to monodromy eigenvalues [2401.17772].

Poles are defined by writing \(T=\mathbb L^{-s}\). The snc formula gives candidate poles
\[
s=-\nu_j/N_j.
\]
The largest candidate
\[
s_{\max}=\max_j(-\nu_j/N_j)
\]
is always an actual pole. The monodromy property is formulated using the ring
\[
\mathrm{Mon}_X:=\mathcal M_k^{\hat\mu}\big[T,(1-\mathbb L^aT^b)^{-1}\big]_{(a,b)\in E_X},
\]
where \(E_X\) consists of pairs \((a,b)\) with \(b>0\) and \(\exp(2\pi ia/b)\) a monodromy eigenvalue. The condition \(Z_{X,\omega}(T)\in \mathrm{Mon}_X\) is independent of \(\omega\). Moreover, if \(X\) and \(X'\) are birational \(K\)-trivial varieties, then \(Z_{X,\omega}(T)\) lies in \(\mathrm{Mon}_X[(1-\mathbb L^{-m})^{-1}]_{m>0}\) if and only if \(Z_{X',\omega'}(T)\) does. Birational invariance also holds for the monodromy eigenvalues themselves, through equality of the polynomials \(P_{X,m}(u)\) and \(P_{X',m}(u)\) for all \(m\ge 0\) [2401.17772].

Known instances of the monodromy property include abelian varieties, Galois-equivariant Kulikov models, certain K3 surfaces, and products. For K3 surfaces with Kulikov models, \(Z\) has a unique pole, and the largest pole always gives a monodromy eigenvalue in middle degree.

## 4. Minimal-model birational zeta functions for divisors

For a smooth \(k\)-variety \(X\) of dimension \(d\) and a non-constant regular function \(f\colon X\to \mathbb A^1\) with reduced divisor \(D=V(f)\), the Denef–Loeser motivic zeta function is
\[
Z_f^{\mathrm{mot}}(T)=\sum_{m\ge 1}[X_m(f)]\mathbb L^{-md}T^m
\]
with
\[
X_m(f)=\{\gamma\in L_m(X)\mid \operatorname{ord}_\gamma(f)=m\}.
\]
Equivalently, using arcs \(\mathcal L(X)=L_\infty(X)\),
\[
Z_f^{\mathrm{mot}}(T)=\sum_{m\ge 1}\mu\big(\{\gamma\in \mathcal L(X)\mid \operatorname{ord}_\gamma(f)=m\}\big)T^m.
\]
If \(\pi\colon Y\to X\) is a log resolution with
\[
(f\circ \pi)^{-1}(0)=\sum_{i\in S}N_iE_i,\qquad K_{Y/X}=\sum_{i\in S}(\nu_i-1)E_i,
\]
then
\[
Z_f^{\mathrm{mot}}(T)=\sum_{\emptyset\neq I\subseteq S}[E_I^\circ]\prod_{i\in I}\frac{\mathbb L-1}{\mathbb L^{\nu_i}T^{-N_i}-1}.
\]
The birational zeta function of the 2025 paper replaces Grothendieck classes by birational equivalence classes and replaces log resolutions by dlt modifications or minimal models [2509.03352].

Let \(\mathrm{Bir}_k^d\) be the set of birational equivalence classes of irreducible \(d\)-dimensional \(k\)-varieties, and let
\[
\mathbb Z[\mathrm{Bir}_k]=\bigoplus_d \mathbb Z[\mathrm{Bir}_k^d]
\]
with birational Lefschetz class
\[
\mathbb L_{\mathrm{bir}}:=\{\mathbb A^1\}.
\]
For a smooth quasi-projective \(X/\mathbb C\), a non-zero reduced divisor \(D\), and a dlt modification
\[
\mu\colon (Y,\Delta)\to (X,D),
\]
with \(\Delta=\mu_*^{-1}D+\operatorname{Exc}^1(\mu)=((\mu^*D)_{\mathrm{red}})\), index set \(S\) for the irreducible components \(E_i\) of \(\Delta\), multiplicities \(N_i=\operatorname{ord}_{E_i}(D)\), and
\[
K_{Y/X}=\sum_{i\in S}(\nu_i-1)E_i,
\]
the global birational zeta function is
\[
Z^{\mathrm{bir}}_{X,D}(T):=\sum_{\emptyset\neq I\subseteq S}\{E_I\}\prod_{i\in I}\frac{1}{\mathbb L_{\mathrm{bir}}^{\nu_i}T^{-N_i}-1}
\in \mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket.
\]
For a closed subset \(\Sigma\subseteq \operatorname{Supp}(D)\), if
\[
S_\Sigma=\{i\in S\mid \mu(E_i)\subseteq \Sigma\},
\]
the local version is
\[
Z^{\mathrm{bir}}_{X,D,\Sigma}(T):=\sum_{\substack{\emptyset\neq I\subseteq S\\ I\cap S_\Sigma\neq \emptyset}}\{E_I\}\prod_{i\in I}\frac{1}{\mathbb L_{\mathrm{bir}}^{\nu_i}T^{-N_i}-1}.
\]
This rational expression is independent of the chosen dlt modification and, more generally, depends only on the crepant-birational equivalence class [2509.03352].

The intrinsic interpretation uses contact loci. For \(\Sigma\subseteq \operatorname{Supp}(D)\),
\[
X_m(X,D,\Sigma)=\{\gamma\in L_m(X)\mid \operatorname{ord}_\gamma(D)=m,\ \gamma(0)\in \Sigma\}.
\]
Dlt valuations with \(\operatorname{ord}_E(D)\mid m\) produce distinct irreducible components of \(X_m\). If \(X_m^{\mathrm{dlt}}\) denotes the union of irreducible components produced by dlt \(m\)-valuations, then the contact-loci definition agrees with the dlt-modification formula. There is also a codimension-one expression
\[
Z^{\mathrm{bir}}_{X,D,\Sigma}(T)=\sum_{E\ \mathrm{dlt\ valuation\ centered\ in}\ \Sigma}\frac{\{E\}}{\mathbb L_{\mathrm{bir}}^{\nu_E}T^{-N_E}-1}.
\]
A key limitation is that arbitrary dlt resolutions that are not minimal over \(X\) may introduce spurious poles.

## 5. Poles, comparisons, and representative computations

In both the \(K\)-trivial and divisor settings, the denominator data determine candidate poles of the form
\[
s=-\nu_i/N_i.
\]
For the \(K\)-trivial motivic zeta function, this follows from the snc expression in the equivariant Grothendieck ring, and the largest candidate \(s_{\max}\) is always an actual pole. For the minimal-model birational zeta function, poles are defined using minimal denominator sets arising from dlt modifications in a crepant-birational class; under the rational specialization \(\rho\colon \mathbb Z[\mathrm{Bir}_{\mathbb C}]\to \mathbb Z[t]\), the largest pole of \(Z^{\mathrm{rat}}_{f,x}(T)\) is at
\[
s_0=-\operatorname{lct}_x(X,D),
\]
and its order equals the maximal number of components meeting at a stratum with \(\nu_j/N_j=\operatorname{lct}_x(X,D)\). In particular, \(-\operatorname{lct}_x(X,D)\) is a pole of \(Z^{\mathrm{bir}}_{f,x}(T)\) [2401.17772] [2509.03352].

For local plane curve singularities, the birational and topological pictures align sharply. The 2025 paper proves that the sets of poles of the local birational zeta function and the local topological zeta function coincide “essentially”: \(s_0\) is a pole of \(Z^{\mathrm{top}}_{f,a}(s)\) if and only if it is detected by the minimal dlt model in the birational theory, and the pole orders also agree. Consequently, the local birational monodromy conjecture holds for plane curves. The node
\[
f(x,y)=xy
\]
gives a basic example: the minimal log resolution is the identity, \(N_1=N_2=1\), \(\nu_1=\nu_2=1\), and the local birational zeta function is
\[
Z^{\mathrm{bir}}_{f,a}(T)=\frac{\mathbb L_{\mathrm{bir}}^2}{\mathbb L_{\mathrm{bir}}^2T^{-2}-1},
\]
with a pole at \(s=-1\) of order \(2\). For the cusp
\[
f(x,y)=y^2-x^3,
\]
the candidate poles are \(\{-1,-5/6\}\), coming from denominators \((\mathbb L T^{-1}-1)\) and \((\mathbb L^5T^{-6}-1)\) in the motivic theory and \((\mathbb L_{\mathrm{bir}}T^{-1}-1)\) and \((\mathbb L_{\mathrm{bir}}^5T^{-6}-1)\) in the birational theory; both are actual poles [2509.03352].

On the \(K\)-trivial side, explicit computations likewise show that poles can carry refined birational information. For K3 surfaces with Kulikov models, \(Z\) has a unique pole. Abelian varieties satisfy the monodromy property. If \(S\) is an abelian surface or a K3 satisfying monodromy, then \(\operatorname{Hilb}^n(S)\) also satisfies monodromy, and birational invariance transfers this property to any \(K\)-trivial variety birational to \(\operatorname{Hilb}^n(S)\) [2401.17772].

## 6. Smooth fillings, limitations, and related constructions

A major application of the \(K\)-trivial theory is the obstruction to smooth fillings. Good reduction means the existence of a smooth, proper algebraic space \(\mathcal X/R\) with generic fiber \(X\). Cohomological good reduction means trivial inertia action on \(H^m(X\times_K K^a,\mathbb Q_l)\) for all \(m\) and \(l\). Good reduction implies cohomological good reduction. If \(X\) is birational to a smooth proper \(Y/K\) with \(K_Y\sim 0\) and \(Y\) has good reduction, then for any volume form \(\omega\) on \(X\) there exist \(v\in \mathbb Z\) and \(\alpha\) in the image of the restriction map \(\mathrm{Res}\colon \mathcal M_k\to \mathcal M_k^{\hat\mu}\) such that
\[
Z_{X,\omega}(T)=\frac{\alpha}{1-\mathbb L^{-v}T}
\quad\text{in }R.
\]
Hence \(Z_{X,\omega}(\mathbb L^{-s})\) has a single pole \(s=-v\in \mathbb Z\). Detecting multiple poles therefore obstructs the existence of a smooth filling, even after birational modifications and finite base change [2401.17772].

Two explicit families illustrate this obstruction. For the Cynk–van Straten threefolds, one obtains a smooth proper \(X/K\) of dimension \(3\) with \(K_X\sim 0\), trivial monodromy, and
\[
Z_{X,\omega}(T)= [F_0^\circ]\frac{T}{1-T}
+ [F_1]\frac{\mathbb L^{-1}T}{1-\mathbb L^{-1}T}
+ [F_{0,1}](\mathbb L-1)\frac{\mathbb L^{-1}T^2}{(1-T)(1-\mathbb L^{-1}T)},
\]
whose poles are exactly \(\{-1,0\}\). Using the Euler–Poincaré realization, both are actual poles, so no smooth filling exists over any finite base change, although the monodromy property still holds. For Voisin’s Lefschetz degenerations in even dimension \(d\ge 4\), the motivic zeta function has poles at \(0\) and \(m_i(1-d)/2\), possibly half-integers, implying monodromy of order \(\le 2\) and again ruling out smooth fillings even after finite base change and birational modifications [2401.17772].

The present theories have strict hypotheses. The \(K\)-trivial birational invariance theorem assumes characteristic zero, smooth proper geometrically connected \(K\)-varieties, trivial canonical bundles, and a volume form. The equality \(Z_{X,\omega}(T)=Z_{X',\omega'}(T)\) is proved in the localized ring \(R\); it is plausible, but currently unknown, that equality already holds in \(\mathcal M_k^{\hat\mu}[T]\). The full monodromy conjecture in this setting remains open in general. For nontrivial canonical bundle, the specific birational invariance statements do not apply. In the divisor setting, the theory is developed over \(\mathbb C\) for smooth quasi-projective \(X\) and non-zero reduced divisors \(D\), and non-minimal dlt resolutions may create extra poles [2401.17772] [2509.03352].

A related but distinct birationally invariant construction is the motivic infinite cyclic zeta function of an SNC divisor \(E\subset X\) with holonomy \(\Delta\) and discrepancy data \(\nu\):
\[
Z_{X,E,\Delta,\nu}^A(T)
=
\sum_{\substack{\emptyset\neq I\subseteq S\\ I\cap A\neq \emptyset}}
[\widetilde{E_I^\circ}](\mathbb L-1)^{|I|-1}
\prod_{i\in I}\frac{\mathbb L^{-\nu_i}T^{m_i}}{1-\mathbb L^{-\nu_i}T^{m_i}}.
\]
It is invariant under the equivalence relation generated by birational maps identifying punctured neighborhoods and transporting holonomy and log discrepancy data. Its limit as \(T\to +\infty\) recovers the motivic infinite cyclic cover, and under log resolution hypotheses it recovers the Denef–Loeser local zeta function of a hypersurface germ. This suggests that birational zeta phenomena form a broader family of invariants linking degenerations, arc spaces, dlt geometry, and monodromy [1702.06590].

Source: https://www.emergentmind.com/topics/birational-zeta-function