Birational Zeta Function: Invariants & Monodromy

• Birational Zeta Function is a birationally invariant series that encodes degeneration, monodromy, and birational data using motivic and divisor frameworks.
• It is defined in multiple settings—such as K-trivial varieties and reduced divisors—by leveraging snc-models, dlt modifications, and localized Grothendieck rings.
• The invariant facilitates the analysis of poles, log discrepancies, and obstructions to smooth fillings through explicit computations and monodromy eigenvalue assessments.

Searching arXiv for the cited papers on birational zeta functions and related motivic zeta constructions. Search query: ([2401.17772](/papers/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 (Lunardon et al., 2024, Biesbrouck et al., 3 Sep 2025, Villa et al., 2017).

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 $K$0 on a smooth complex quasi-projective variety, with holonomy $K$1 and discrepancies $K$2	$K$3

In the $K$4-trivial setting, the series measures how $K$5 degenerates at $K$6, and the use of $K$7-equivariant coefficients records monodromy through $K$8-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 (Lunardon et al., 2024, Biesbrouck et al., 3 Sep 2025, Villa et al., 2017).

2. Motivic zeta functions of $K$9-trivial varieties

Let $K$0 be a field of characteristic zero, $K$1, and $K$2. The profinite group scheme of roots of unity is $K$3. The relevant coefficient ring is the equivariant Grothendieck ring $K$4, localized by the Lefschetz motive $K$5, giving $K$6. For technical birational statements one further localizes to

$K$7

If $K$8 is smooth, proper, geometrically connected, of pure dimension $K$9, with trivial canonical bundle $K_X \sim 0$0, and $K_X \sim 0$1 is a volume form on $K_X \sim 0$2, then the motivic zeta function is

$K_X \sim 0$3

where $K_X \sim 0$4. If $K_X \sim 0$5, then

$K_X \sim 0$6

Thus rescaling the volume form shifts the $K_X \sim 0$7-variable by a power of $K_X \sim 0$8 (Lunardon et al., 2024).

The rational form is obtained from an snc-model $K_X \sim 0$9 of $\omega$0, with special fiber

$\omega$1

For $\omega$2, one sets

$\omega$3

and $\omega$4. If $\omega$5 denotes the pullback of $\omega$6 to the normalization of $\omega$7, then $\omega$8 is a $\omega$9-torsor. Viewing $\mathcal M_k^{\hat\mu}$0 as a rational section of the logarithmic relative canonical bundle, one writes

$\mathcal M_k^{\hat\mu}$1

The Denef–Loeser-style formula is

$\mathcal M_k^{\hat\mu}$2

This exhibits rationality and makes the candidate poles visible through the denominators $\mathcal M_k^{\hat\mu}$3 (Lunardon et al., 2024).

The same paper defines a generalized zeta function $\mathcal M_k^{\hat\mu}$4 for germs $\mathcal M_k^{\hat\mu}$5 of canonical forms on function fields $\mathcal M_k^{\hat\mu}$6 and proves that it is independent of the chosen snc-model. For smooth proper $\mathcal M_k^{\hat\mu}$7-trivial $\mathcal M_k^{\hat\mu}$8, this generalized construction reduces to the original $\mathcal M_k^{\hat\mu}$9.

3. Birational invariance and monodromy in the $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$0-trivial setting

The central birational invariance statement is the following. If $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$1 and $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$2 are smooth, proper, geometrically connected $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$3-schemes with $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$4, and $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$5 is birational, then for a volume form $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$6 on $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$7 and the uniquely induced volume form $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$8 on $$R=\mathcal M_k^{\hat\mu}[T,(1-\mathbb L^{-m})^{-1}]_{m>0}$$9, one has

$X/\mathbb C$0

The proof proceeds by constructing $X/\mathbb C$1 for germs $X/\mathbb C$2 and proving independence of snc-model via weak factorization tailored to $X/\mathbb C$3-elementary blow-ups. This bypasses the lack of a change-of-variables formula for motivic integrals of canonical forms over $X/\mathbb C$4 (Lunardon et al., 2024).

The same framework organizes the monodromy problem. For a smooth proper $X/\mathbb C$5, the $X/\mathbb C$6-adic characteristic polynomial

$X/\mathbb C$7

is cyclotomic and independent of $X/\mathbb C$8 and of the chosen topological generator $X/\mathbb C$9; it is denoted $D$0. For an snc-model with special fiber $D$1, the A’Campo-type formula is

$D$2

This ties multiplicities $D$3 directly to monodromy eigenvalues (Lunardon et al., 2024).

Poles are defined by writing $D$4. The snc formula gives candidate poles

$D$5

The largest candidate

$D$6

is always an actual pole. The monodromy property is formulated using the ring

$D$7

where $D$8 consists of pairs $D$9 with $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$0 and $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$1 a monodromy eigenvalue. The condition $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$2 is independent of $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$3. Moreover, if $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$4 and $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$5 are birational $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$6-trivial varieties, then $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$7 lies in $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$8 if and only if $$\mathbb Z[\mathrm{Bir}_{\mathbb C}][\mathbb L_{\mathrm{bir}}^{-1}]\llbracket T\rrbracket$$9 does. Birational invariance also holds for the monodromy eigenvalues themselves, through equality of the polynomials $K$00 and $K$01 for all $K$02 (Lunardon et al., 2024).

Known instances of the monodromy property include abelian varieties, Galois-equivariant Kulikov models, certain K3 surfaces, and products. For K3 surfaces with Kulikov models, $K$03 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$04-variety $K$05 of dimension $K$06 and a non-constant regular function $K$07 with reduced divisor $K$08, the Denef–Loeser motivic zeta function is

$K$09

with

$K$10

Equivalently, using arcs $K$11,

$K$12

If $K$13 is a log resolution with

$K$14

then

$K$15

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 (Biesbrouck et al., 3 Sep 2025).

Let $K$16 be the set of birational equivalence classes of irreducible $K$17-dimensional $K$18-varieties, and let

$K$19

with birational Lefschetz class

$K$20

For a smooth quasi-projective $K$21, a non-zero reduced divisor $K$22, and a dlt modification

$K$23

with $K$24, index set $K$25 for the irreducible components $K$26 of $K$27, multiplicities $K$28, and

$K$29

the global birational zeta function is

$K$30

For a closed subset $K$31, if

$K$32

the local version is

$K$33

This rational expression is independent of the chosen dlt modification and, more generally, depends only on the crepant-birational equivalence class (Biesbrouck et al., 3 Sep 2025).

The intrinsic interpretation uses contact loci. For $K$34,

$K$35

Dlt valuations with $K$36 produce distinct irreducible components of $K$37. If $K$38 denotes the union of irreducible components produced by dlt $K$39-valuations, then the contact-loci definition agrees with the dlt-modification formula. There is also a codimension-one expression

$K$40

A key limitation is that arbitrary dlt resolutions that are not minimal over $K$41 may introduce spurious poles.

5. Poles, comparisons, and representative computations

In both the $K$42-trivial and divisor settings, the denominator data determine candidate poles of the form

$K$43

For the $K$44-trivial motivic zeta function, this follows from the snc expression in the equivariant Grothendieck ring, and the largest candidate $K$45 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 $K$46, the largest pole of $K$47 is at

$K$48

and its order equals the maximal number of components meeting at a stratum with $K$49. In particular, $K$50 is a pole of $K$51 (Lunardon et al., 2024, Biesbrouck et al., 3 Sep 2025).

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”: $K$52 is a pole of $K$53 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

$K$54

gives a basic example: the minimal log resolution is the identity, $K$55, $K$56, and the local birational zeta function is

$K$57

with a pole at $K$58 of order $K$59. For the cusp

$K$60

the candidate poles are $K$61, coming from denominators $K$62 and $K$63 in the motivic theory and $K$64 and $K$65 in the birational theory; both are actual poles (Biesbrouck et al., 3 Sep 2025).

On the $K$66-trivial side, explicit computations likewise show that poles can carry refined birational information. For K3 surfaces with Kulikov models, $K$67 has a unique pole. Abelian varieties satisfy the monodromy property. If $K$68 is an abelian surface or a K3 satisfying monodromy, then $K$69 also satisfies monodromy, and birational invariance transfers this property to any $K$70-trivial variety birational to $K$71 (Lunardon et al., 2024).

6. Smooth fillings, limitations, and related constructions

A major application of the $K$72-trivial theory is the obstruction to smooth fillings. Good reduction means the existence of a smooth, proper algebraic space $K$73 with generic fiber $K$74. Cohomological good reduction means trivial inertia action on $K$75 for all $K$76 and $K$77. Good reduction implies cohomological good reduction. If $K$78 is birational to a smooth proper $K$79 with $K$80 and $K$81 has good reduction, then for any volume form $K$82 on $K$83 there exist $K$84 and $K$85 in the image of the restriction map $K$86 such that

$K$87

Hence $K$88 has a single pole $K$89. Detecting multiple poles therefore obstructs the existence of a smooth filling, even after birational modifications and finite base change (Lunardon et al., 2024).

Two explicit families illustrate this obstruction. For the Cynk–van Straten threefolds, one obtains a smooth proper $K$90 of dimension $K$91 with $K$92, trivial monodromy, and

$K$93

whose poles are exactly $K$94. 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 $K$95, the motivic zeta function has poles at $K$96 and $K$97, possibly half-integers, implying monodromy of order $K$98 and again ruling out smooth fillings even after finite base change and birational modifications (Lunardon et al., 2024).

The present theories have strict hypotheses. The $K$99-trivial birational invariance theorem assumes characteristic zero, smooth proper geometrically connected $K$00-varieties, trivial canonical bundles, and a volume form. The equality $K$01 is proved in the localized ring $K$02; it is plausible, but currently unknown, that equality already holds in $K$03. 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 $K$04 for smooth quasi-projective $K$05 and non-zero reduced divisors $K$06, and non-minimal dlt resolutions may create extra poles (Lunardon et al., 2024, Biesbrouck et al., 3 Sep 2025).

A related but distinct birationally invariant construction is the motivic infinite cyclic zeta function of an SNC divisor $K$07 with holonomy $K$08 and discrepancy data $K$09: $K$10 It is invariant under the equivalence relation generated by birational maps identifying punctured neighborhoods and transporting holonomy and log discrepancy data. Its limit as $K$11 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 (Villa et al., 2017).