Motivic Monodromy in Degenerating Varieties

• Motivic monodromy is a framework linking degenerating algebraic varieties to the action of monodromy or Galois groups, providing a motivic and birationally invariant perspective.
• It employs techniques such as motivic integration, monodromy pairings, and limit mixed Hodge structures to compute cohomological and arithmetic invariants.
• The theory produces explicit formulas for invariants in settings like maximally degenerate K3 and Calabi–Yau varieties, ensuring birational invariance and practical computation.

Motivic monodromy is a concept linking the structure of degenerating algebraic varieties over local or non-archimedean fields with the action of the Galois or monodromy group on their cohomology. It encodes—in a motivic, birationally invariant, and often functorial way—the interplay between arithmetic, geometry, and topology in the behavior of degenerations. The subject spans several domains: motivic integration, monodromy pairings, weight filtrations, limit mixed Hodge structures, and Hodge-theoretic and $\ell$-adic invariants. Central objects include the motivic monodromy pairing and the motivic monodromy conjecture, as well as refined formulas such as the motivic integral and the discriminant invariants arising from cohomological pairings.

1. Motivic Integration and the Motivic Integral

Let $R$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ is generated by types of $k$-varieties up to scissors relations, localized by inverting the class $\mathbb{L} = [\mathbb{A}^1_k]$. The motivic measure is taken in $$K_0(\mathrm{Var}_k)_{\mathrm{loc}} = K_0(\mathrm{Var}_k)[\mathbb{L}^{-1}]$$, with "Tate twists" $[X](n) := [X] \cdot \mathbb{L}^{-n}$ encoding shifts in cohomological weight.

For a smooth, proper, connected, Calabi–Yau $K$-variety $X$ (i.e., $\omega_X$ trivial), a weak Néron model $V/R$ is fixed, together with a nowhere-vanishing top-degree form $\omega\in H^0(X,\omega_X)$. In the special fiber $V_0 = \cup_i V_i$, the divisor of $\omega$ decomposes as $\operatorname{div}\omega = \sum_i m_i V_i$. The motivic integral is defined as:

$$\int_X := \sum_i [V_i^\circ](m_i - \min_j m_j) \in K_0(\mathrm{Var}_k)_{\mathrm{loc}}$$

where $V_i^\circ$ is the smooth locus of $V_i$. This invariant is independent of the choice of $V$ or $\omega$ and specializes to $p$-adic volume over finite fields after taking point-counts $[Z]\mapsto |Z(\mathbb{F}_q)|$.

2. The Motivic Monodromy Pairing

Given a smooth $K$-variety $X$ of dimension $d$, let $\bar{K}$ be the completion of an algebraic closure of $K$ with Galois group $G = \mathrm{Gal}(\bar{K}/K)$. For any prime $\ell\neq \mathrm{char}\,k$, the classical monodromy operator $N = \log$(unipotent part of monodromy) acts on $H^m(X_{\bar{K}},\mathbb{Q}_\ell)$. Define:

$$T_\ell^m(X) := \mathrm{Im}\left( H^m(X_{\bar{K}},\mathbb{Q}_\ell)(m) \xrightarrow{N^m} H^m(X_{\bar{K}},\mathbb{Q}_\ell)\right)$$

and for $m = d$, write $T_\ell^d(X) = N^d H^d(X_{\bar{K}},\mathbb{Q}_\ell)$. Theorem 3 of (Stewart et al., 2011) provides a canonical identification

$$T_\ell^d(X) \simeq H^d_{\mathrm{sing}}(|X^{\mathrm{an}}_{\bar{K}}|, \mathbb{Q}_\ell)$$

independent of $\ell$.

A positive-definite, non-degenerate pairing is constructed:

$$(-,-)_\ell: T_\ell^d(X) \otimes T_\ell^d(X) \to \mathbb{Q}_\ell$$

which, after extension to $\mathbb{Q}$ and passage to the limit, defines the motivic monodromy pairing

$$(-,-)_\mathrm{mot}: T^d(X) \otimes T^d(X) \to \mathbb{Q}$$

where $T^d(X)$ is the $\mathbb{Q}$-lattice associated to the image of $N^d$ in cohomology (via alternation and transfer). The discriminant of the pairing, $$r^d(X,K) := \mathrm{Disc}((-,-)_\mathrm{mot})^{-1} \in \mathbb{Q}_{>0}$$, yields a "monodromy invariant" which is birational under base-change.

3. Birational Invariance and Étale Descent

The functors $X\mapsto T_\ell^d(X)$ and $X\mapsto \mathrm{Im}(N^d)$ admit transfer maps for finite morphisms, and both are unchanged under replacing $X$ by an open dense $U \subset X$ (see Theorems and Lemmas in §4 of (Stewart et al., 2011)). De Jong’s alterations theorem produces alterations $X'\to X$ admitting strictly semi-stable projective models, enabling comparison of $T^d$ and the monodromy pairing at the level of birational models. This yields the strong birational invariance result for the motivic monodromy pairing.

4. Relation to Limit Mixed Hodge Structures

When $K=\mathbb{C}((t))$ and $X/K$ is projective, the associated limit mixed Hodge structure is

$$H^m(\lim X) = (H^m(\lim X, \mathbb{Z}),\, W_\bullet,\, F^\bullet)$$

arising from the log-geometry extension of the Schmid–Steenbrink theory (cf. Schmid–Steenbrink, logarithmic degenerations). The weight spectral sequence

$$E_1^{p,q} = \bigoplus_{i-p\geq 0} H^{q + 2p - 2i}(Y^{(2i-p)},\mathbb{Z})(p-i)$$

with $Y$ the special fiber of a semi-stable model, degenerates at $E_2$, and the monodromy operator acts with $N^n: \mathrm{Gr}_{m+n}^W \to \mathrm{Gr}_{m-n}^W$ an isomorphism (Deligne).

There is a canonical isomorphism (Remark 4.3 in (Stewart et al., 2011))

$$T^d(X) \otimes \mathbb{Q} \xrightarrow{\sim} W_0 H^d(\lim X, \mathbb{Q})$$

under which the motivic monodromy pairing corresponds to the polarized pairing on the lowest weight piece, with formula:

$$(x,y) \mapsto (-1)^{d(d-1)/2} \langle x, N^{d-1} y' \rangle$$

with $y'$ a lift under $N^d$ (see Remark 4.6 and formula (4.7) in (Stewart et al., 2011)).

5. Explicit Formulas for Maximally Degenerate K3 Surfaces

A projective Calabi–Yau $X/K$ is maximally degenerate if $T^d(X)\neq 0$ (i.e., $\mathrm{Im}(N^d)\neq 0$). The conjectural formula (Conjecture in §5 of (Stewart et al., 2011)) for a maximally degenerate K3 surface posits:

• $|X^{\mathrm{an}}_{\bar{K}}| \simeq S^2$; thus $T^d(X) \cong \mathbb{Z}$,
• The lattice $$T^d_\ell(X) \subset H^d(X_{\bar{K}},\mathbb{Z}_\ell)$$ is saturated for each $\ell$,
• There exists $K'\supset K$ and $r=r_2(X,K)\in \mathbb{Z}_+$ such that for every finite extension $L/K'$ of ramification index $e$:

$$\int_{X_L} = (2r+2)\, Q(0) + (20 - e^2 r^2)\, Q(-1) + e^2 r^2\, Q(-2)$$

in the localized Grothendieck ring, with $Q(n) := \mathbb{L}^{-n} \otimes \mathbb{Q}$.

In the Kummer case ($\mathrm{char}\,k\neq 2$), with $X$ associated to an abelian surface $A/K$ admitting split semi-stable reduction, the canonical polystable formal model is constructed via Mumford’s model, appropriate blow-ups, and involution quotient. The explicit calculations match the formula, and $$T^d(X)\cong \Lambda/2\Lambda \cong \mathbb{Z}/r\mathbb{Z}$$, with the smooth locus of the special fiber contributing the precise motivic integral coefficients.

6. Broader Context: Relation to Other Motivic Monodromy Theories

The motivic monodromy pairing and monodromy invariants generalize classical monodromy pairings for abelian varieties (Grothendieck) and are related to limit mixed Hodge structures and their polarizations. The construction is naturally compatible with the birational and cohomological invariance properties desirable for arithmetic and birational geometry. The invariants provide structural bridges between the combinatorics of special fiber degenerations, the topology of Berkovich analytifications, and the Hodge/arithmetic monodromy data. These techniques feed directly into deeper conjectures on the structure of motivic zeta functions and their role as obstructions to smooth or semi-stable fillings (Lunardon et al., 31 Jan 2024).

7. Significance and Applications

Motivic monodromy provides a precise link between the degenerations of Calabi–Yau and K-trivial varieties over non-archimedean fields and their cohomological and Hodge-theoretic invariants. For maximally degenerate K3 surfaces, the explicit motivic formulae relate integral data on special fiber components to monodromy and limit Hodge structures, resolving questions on the realization of monodromy in birationally meaningful motivic terms. The birational invariance and explicit models over formal schemes facilitate computations in non-archimedean geometry and underlie the extension of the monodromy conjecture to broader K-trivial and Calabi–Yau settings.

These results demonstrate deep cohomological, arithmetic, and motivic compatibility, unifying several perspectives: strictly birational invariance, monodromy actions in étale and Hodge cohomology, and explicit motivic invariants constructed by integration over models. The techniques deployed, involving alterations, semi-stable models, and limit mixed Hodge structures, are central in modern approaches to the paper of arithmetic and geometry of degenerations.