Transcendental b-divisors II -- Monotonicity theorem

1. Problem Statement and Motivation

The paper addresses a fundamental problem in complex geometry and pluripotential theory: quantifying the loss of mass (volume) when the singularities of positive $(1,1)$-currents increase.

• Context: Let $X$ be a compact Kähler manifold. Given two closed positive $(1,1)$-currents $T$ and $T'$ in the same cohomology class where $T$ is "more singular" than $T'$ (denoted $T \preceq_I T'$, meaning the multiplier ideal sheaves of $T$ are contained in those of $T'$, or equivalently, the Lelong numbers of $T$ dominate those of $T'$), the monotonicity theorem (Witt Nyström, Darvas-Demailly-Nguyen-Lu) states that the volume of the less singular current is larger: $\text{vol}(T') \geq \text{vol}(T)$.
• The Gap: While the inequality is known, previous works (e.g., by Vu, Su) provided only non-optimal lower bounds for the difference $\text{vol}(T') - \text{vol}(T)$ in terms of the differences in Lelong numbers. These bounds often lacked explicit constants or precise structural descriptions of the "lost mass."
• Goal: The author aims to provide a quantitative and optimal control of the mass loss, expressing the difference explicitly in terms of the differences in Lelong numbers and restricted volumes, utilizing the theory of nef b-divisors.

2. Methodology

The paper relies heavily on the extension of intersection theory to transcendental b-divisors, a framework developed in the author's previous work [Xia25b] and related literature [DF22].

• Nef B-Divisors: The author treats b-divisors not just as algebraic objects but as families of cohomology classes over all modifications (blow-ups) of $X$. A nef b-divisor is a compatible family of movable classes.
• Generalized Intersection Theory: The core methodological innovation is the development of a linear intersection theory for nef b-divisors. Unlike the classical movable intersection product (which is non-linear and defined via limits), the intersection of nef b-divisors is linear and well-defined on the space of b-classes.
• Restricted Volumes: The author defines a restricted volume for b-divisors, denoted $\text{vol}_{X|D}(D_1, \dots, D_p)$, which generalizes the restricted volume of cohomology classes. This measures the "volume" of the intersection restricted to a prime divisor $D$.
• Trace Operators: The paper utilizes trace operators to relate b-divisors on $X$ to b-divisors on the normalization of a divisor $D$, allowing the decomposition of global intersection numbers into contributions from singularities.
• Okounkov Bodies: In the analysis of mass loss, the author employs transcendental Okounkov bodies (recently developed in [DRWN+26]) to establish sharp estimates for restricted volumes, bridging pluripotential theory with convex geometry.

3. Key Contributions and Results

A. General Intersection Theory of Nef B-Divisors (Theorem A)

The paper establishes a robust intersection theory for nef b-divisors $D_1, \dots, D_p$:

1. Linearity: The map $(D_1, \dots, D_p) \mapsto D_1 \cap \dots \cap D_p$ is multi-linear (additive and homogeneous for non-negative scalars).
2. Relation to Movable Intersection: If $D_i = D(\alpha_i)$ are b-divisors induced by pseudoeffective classes $\alpha_i$, the b-divisor intersection recovers the classical movable intersection $\langle \alpha_1 \wedge \dots \wedge \alpha_p \rangle$ as a limit over modifications.
3. Relation to Non-Pluripolar Products: If $D_i = D(T_i)$ are induced by currents $T_i$, the intersection recovers the cohomology class of the non-pluripolar product $\{T_1 \wedge \dots \wedge T_p\}$.

B. The Quantitative Monotonicity Theorem (Theorem B)

This is the central result. It provides a precise inequality for the difference in intersection products when singularities increase.

• Statement: Let $D_1, \dots, D_p$ be nef b-divisors and $T, T'$ be currents in the same class with $T \preceq_I T'$. Then:
  $$(D_1 \cap \dots \cap D_p \cap D(T')) - (D_1 \cap \dots \cap D_p \cap D(T)) \geq \sum_{D/X} (\nu(T, D) - \nu(T', D)) \cdot \text{vol}_{X|D}(D_1, \dots, D_p)$$
  where the sum runs over all prime divisors $D$ over $X$, and $\nu(T, D)$ is the generic Lelong number.
• Significance: This inequality is optimal. When the $D_i$ are Cartier, the inequality becomes an equality. It decomposes the global mass loss into a sum of local contributions weighted by the restricted volumes.

C. Explicit Mass Loss Estimates (Theorem C & Corollaries)

The author derives explicit lower bounds for the volume difference $\text{vol}(S) - \text{vol}(T)$ in terms of Lelong numbers and the "width" of the cohomology class.

• Theorem C: For a big class $\alpha$ and currents $S, T \in \alpha$ with $T \preceq_I S$:
  $$\text{vol}(S) - \text{vol}(T) \geq (\nu(T, D) - \nu(S, D))^n \cdot \frac{\text{vol}(S)}{(\nu_{\max}(\alpha, D) - \nu(\alpha, D))^n}$$
  Here, $\nu_{\max}(\alpha, D)$ is the maximal Lelong number in the class.
• Improvement: This improves upon previous results by Vu and Su by providing explicit constants and showing that the constant depends on the geometry of the class (specifically the "width" of the class along the divisor), rather than being a universal constant.

D. Connection to Convex Geometry (Theorem D & Corollary 1.5)

In the toric setting, the author demonstrates that the limit measure appearing in the monotonicity theorem corresponds to the mixed area measure of convex bodies.

• Result: The formula recovers the classical Minkowski volume formula:
  $$\text{vol}(P_1, \dots, P_{n-1}, Q') - \text{vol}(P_1, \dots, P_{n-1}, Q) = \frac{1}{n} \int_{S^{n-1}} (\text{Supp}_{Q'} - \text{Supp}_Q) \, dS(P_1, \dots, P_{n-1})$$
  This establishes a deep link between the transcendental monotonicity theorem and classical convex geometry.

E. Equality Cases and Morse Inequalities

The paper discusses conditions under which the inequality becomes an equality.

• Projective Manifolds: Assuming the transcendental Morse inequality (proven for projective manifolds by Witt Nyström), the author proves that the monotonicity formula holds as an equality for all nef b-divisors on projective manifolds.
• Dimension 1 & 2: The equality holds unconditionally for Riemann surfaces (dimension 1) and Kähler surfaces (dimension 2, due to Deng's thesis).

4. Significance and Impact

1. Optimality: The paper provides the first optimal quantitative estimates for mass loss in pluripotential theory, resolving the issue of non-optimal constants in previous literature.
2. Unification: It unifies the theory of movable intersections, non-pluripolar products, and b-divisors into a single linear framework, simplifying complex calculations involving singularities.
3. New Tools: The introduction of restricted volumes for b-divisors offers a new tool for studying the geometry of cohomology classes and their singularities.
4. Bridge to Convex Geometry: By recovering the Minkowski volume formula in the toric case, the paper suggests that the "limit measures" in the transcendental setting are natural generalizations of mixed area measures, opening new avenues for studying dynamical systems and Kähler geometry via convex geometric intuition.
5. Explicit Constants: The explicit dependence of the mass loss on the "width" of the class ($\nu_{\max} - \nu_{\min}$) provides a more nuanced understanding of how the geometry of the class constrains the possible singularities of currents within it.

In summary, Mingchen Xia's paper significantly advances the intersection theory of transcendental currents by leveraging the power of b-divisors to derive sharp, quantitative monotonicity theorems, thereby deepening the connection between complex geometry, pluripotential theory, and convex geometry.