Invariance of Relative Log Canonical Volumes

• The paper demonstrates that the relative log canonical volume remains constant under deformations for KLT fiber pairs, ensuring invariant pluricanonical measures in smooth loci.
• It employs analytic techniques such as AZD, Bergman kernels, and the Ohsawa–Takegoshi extension theorem to establish local constancy of plurigenera and volumes.
• The findings have significant implications for moduli theory and the minimal model program by guaranteeing stability of key numerical invariants in families.

Deformation invariance of relative log canonical volumes is a foundational property in higher-dimensional birational geometry, asserting that the asymptotic measure of the pluricanonical systems of fibers in certain families remains constant as the complex structure deforms. Specifically, for a projective morphism $f: X \to Y$ of complex manifolds (or varieties) equipped with an effective $\mathbb{Q}$-divisor $\Delta$, the relative log canonical volume of the fibers $X_y$ with respect to $K_{X/Y}+\Delta$ is independent of $y$ within smooth loci where the pair $(X_y, \Delta_{X_y})$ has mild singularities—most notably, is Kawamata–log-terminal (KLT) or log canonical. This result and its generalizations, coupled with associated invariance of plurigenera, underpin major advances in the minimal model program and moduli theory for higher-dimensional varieties and pairs (Tsuji, 2010, Fan, 26 Dec 2025).

1. Statement of the Deformation Invariance Property

Let $f: X \to Y$ be a surjective, projective morphism of complex manifolds with $n$-dimensional connected fibers, and let $\Delta = \sum_i d_i D_i$ ($0 \leq d_i < 1$) be an effective $\mathbb{Q}$-divisor such that $(X, \Delta)$ is KLT on each smooth fiber. The relative log canonical line bundle is defined as

$$K_{X/Y} + \Delta = K_X \otimes f^* K_Y^{-1} \otimes \mathcal{O}_X(\Delta) .$$

For each $y \in Y$, and each $m \geq 1$ such that $m\Delta$ is integral, define the $m$th logarithmic plurigenus

$$P_m(y) = \dim H^0(X_y, \mathcal{O}_{X_y}(\lfloor m(K_{X/Y} + \Delta) \rfloor))$$

and the fiberwise log canonical volume

$$\operatorname{vol}_{X_y}(K_{X/Y} + \Delta) := \limsup_{m \to \infty} \frac{P_m(y)}{m^n/n!}.$$

Theorem (Invariance of Volumes): Under the above assumptions, the function

$$y \mapsto \operatorname{vol}_{X_y}(K_{X/Y} + \Delta)$$

is locally constant on the locus of $Y$ where $f$ is smooth and $(X_y, \Delta|_{X_y})$ is KLT (Tsuji, 2010). This result extends, under suitable hypotheses, to families with log canonical fibers and in weak semistable families, to invariance of fiberwise relative log canonical volumes (Fan, 26 Dec 2025).

2. Technical Framework and Definitions

Key definitions are crucial for precise formulation and application:

• KLT pair: $(X, \Delta)$ is Kawamata–log-terminal if $X$ is normal, $\Delta$ is effective with coefficients $<1$, and discrepancies $a_i > -1$ for all exceptional divisors $E_i$ in any log resolution $p: Y \to X$ so that $K_Y + \Delta_Y = p^* (K_X + \Delta) + \sum a_i E_i$.
• Relative log canonical divisor: For $f: X \to T$, $K_{X/T} = K_X - f^* K_T$, with $K_{X/T} + B$ for a divisor $B$.
• Relative log canonical volume: For each fiber $X_t$,

$$\operatorname{vol}(X_t, (K_{X/T}+B)_t) = \limsup_{m \to \infty} \frac{n! \cdot h^0(X_t, \lfloor m(K_{X/T}+B)_t \rfloor)}{m^n}$$

where $n = \dim X_t$, $h^0$ denotes the space of global sections, and $\lfloor\cdot\rfloor$ the round-down (Fan, 26 Dec 2025).

• Weak semistable morphisms: Projective morphisms with connected, reduced fibers where the domain and target are toroidal pairs, the morphism is flat, and the combinatorics are controlled locally by toric models (Fan, 26 Dec 2025).
• Singular Hermitian metric: A metric $h$ on a line bundle $L$ is given locally by $\|e\|_h^2 = e^{-\varphi}$, with $\varphi$ upper-semicontinuous; curvature $\Theta_h = i \partial \bar\partial \varphi$.
• Multiplier ideals $\mathcal{I}(h)$: Consist locally of holomorphic functions $f$ such that $|f|^2 e^{-\varphi}$ is $L^1$.

3. Methods and Proof Structures

The rigidity of log canonical volumes under deformation is established via analytic and algebro-geometric techniques:

• Construction of canonical singular Hermitian metrics: The central tool is the analytic Zariski decomposition (AZD), denoted "supercanonical" metric $h_{\mathrm{can}}$ on $K_{X/Y} + \Delta$, uniquely determined by the fibration and divisor. Its curvature current is semipositive, and it yields on smooth fibers the AZD with minimal singularities. The metric is constructed using Bergman kernels associated with ample line bundles perturbed by multiples of the canonical bundle and divisor. Plurisubharmonic variation and Ohsawa–Takegoshi $L^2$-extension theorems ensure extension properties across fibers (Tsuji, 2010).
• Toroidal and weak semistable reduction: In the context of weak semistable families, toroidal geometry is systematically used to control extractions and to ensure irreducibility of strata on fibers. Appropriate finite base changes and toroidal blowups allow reduction to KLT or even canonical models, preserving relevant volume properties (Fan, 26 Dec 2025).
• Vanishing and base-change theorems: Kawamata–Viehweg vanishing, semicontinuity of cohomology, and properties of direct images under base change guarantee the local constancy of plurigenera and volumes across the base (Tsuji, 2010, Fan, 26 Dec 2025).
• MMP and Birkar–Cascini–Hacon–McKernan techniques: To address pseudo-effectivity and adjoint divisor constructions, perturbations with ample divisors and MMP over the base are deployed, ensuring bigness and invariance of global sections in the linear series of canonical type divisors (Fan, 26 Dec 2025).

4. Generalizations and Related Theorems

Two critical extensions include:

• Twisted invariance: The volume and plurigenera invariance generalize to "twisted" settings where one replaces $\Delta$ by an arbitrary $\mathbb{Q}$-line bundle $B$ with a KLT singular Hermitian metric $h_B$; the same local constancy of fiberwise dimensions holds (Tsuji, 2010).
• Weak semistable and log canonical settings: For weak semistable families as in (Fan, 26 Dec 2025), the hypothesis may be weakened from KLT to log canonical. The deformation invariance theorem holds on the base smooth locus, and in particular encompasses families of foliations birationally bounded by certain algebraically integrable families, yielding the DCC for their set of volumes.

Examples:

Case	Structure on Fibers	Invariance Statement
$\Delta = 0$	Smooth projective varieties	Invariance of plurigenera (Siu, Tsuji)
$\Delta$ normal crossing	Log smooth pairs	Invariance of log plurigenera
$B$ KLT $\mathbb{Q}$-divisor	KLT with singular Hermitian	Twisted invariance (Thm. 1.19, Tsuji)
Weak semistable, log canonical	Canonical or log canonical	Volume invariance (Thm. 4.7, (Fan, 26 Dec 2025))

5. Significance and Applications in Birational Geometry

The deformation invariance of relative log canonical volumes is pivotal in:

• Moduli theory: It enables the construction of well-behaved moduli spaces for varieties or pairs of general type, where invariants remain stable in families.
• Minimal model program: Many structural results, such as the DCC for the set of log canonical volumes and the construction of canonical models, utilize invariance properties to control variation across families and restrict pathologies in the birational classification.
• Foliated and higher relative settings: Recent work extends the invariance to foliations (log canonical foliations) and families birationally bounded by algebraically integrable families, giving rise to finiteness and boundedness results for canonical volumes in these broader contexts (Fan, 26 Dec 2025).

6. Open Problems and Further Directions

A central open question, as highlighted in (Fan, 26 Dec 2025), regards whether $h^0(X_t, m(K_{X_t/Z_t}+B_t))$ is itself independent of $t$ for each fixed divisible $m$ in the general weak semistable, log canonical context. While the volume (asymptotic) invariance is established, the local constancy of the plurigenera remains unproven in full generality when singularities or reducibility are present in the fibers.

The deformation invariance property continues to guide further investigations into the behavior of higher direct images, the stability of numerical invariants in moduli, and the extension of these results to singular base spaces or analytic families.

• H. Tsuji, "Canonical singular hermitian metrics on relative logcanonical bundles" (Tsuji, 2010).
• "Volumes of foliations birationally bounded by algebraically integrable families" (Fan, 26 Dec 2025).
• B. Berndtsson, M. Păun, "Bergman kernels and the pseudoeffectivity of relative canonical bundles".
• T. Ohsawa, K. Takegoshi, "L^2-extension of holomorphic functions".
• C. D. Hacon, J. McKernan, C. Xu, "ACC for log canonical thresholds".