A counterexample to the bounded mass property

Technical Summary: A Counterexample to the Bounded Mass Property

Problem Statement
The paper addresses a fundamental question in pluripotential theory on compact complex manifolds: the validity of the bounded mass property. On a compact Kähler manifold, the total Monge–Ampère mass $\int_X (\omega + dd^c \phi)^n$ is invariant under the potential $\phi$ due to Stokes' theorem. However, on general compact complex manifolds equipped with a Hermitian form $\omega$ (which is not necessarily closed), this mass depends on $\phi$.

The central question, posed by Boucksom–Guedj–Lu [BGL25], is whether the upper Monge–Ampère volume
$$\text{vol}(\omega) := \sup \left\{ \int_X (\omega + dd^c \phi)^n : \phi \in C^\infty(X, \mathbb{R}), \ \omega + dd^c \phi > 0 \right\}$$
is finite for every compact complex manifold. While this property is known to hold for manifolds of dimension $n \le 2$ and those in the Fujiki class, its validity for general manifolds, particularly Hopf manifolds of dimension $n \ge 3$, remained open.

Methodology and Construction
The authors construct a counterexample on the Hopf threefold $X = (\mathbb{C}^3 \setminus \{0\}) / \langle z \mapsto e^{-1}z \rangle$. The proof strategy relies on the geometric structure of $X$ as an elliptic fibration $\pi: X \to \mathbb{P}^2$ with fiber $E = \mathbb{C}/(\mathbb{Z} + 2\pi i \mathbb{Z})$.

1. Background Form and Mass Identity:
   The authors define a specific Hermitian form $\omega = \alpha + \beta$, where $\alpha = \pi^* \omega_{FS}$ is the pullback of the Fubini–Study form and $\beta$ is a rank-one semipositive form positive along fibers. A crucial identity is derived: $dd^c \beta = -\alpha^2$.
   Using Stokes' theorem and the specific cohomological properties of $\omega$, they establish an exact mass identity (Lemma 2.3):
   $$\int_X (\omega + dd^c V)^3 = \int_X \omega^3 + 12\pi \int_{\mathbb{P}^2} \left( \int_{\pi^{-1}(y)} |V_w|^2 \, dA_E \right) \omega_{FS}^2(y),$$
   where $V_w$ denotes the fiberwise derivative and $dA_E$ is the normalized Haar measure on the fiber. This identity reveals that the total mass exceeds the base volume by a term proportional to the fiberwise Dirichlet energy of the potential.

2. Positivity Criterion:
   To ensure the constructed potentials $V$ yield a valid Hermitian form ($\omega + dd^c V > 0$), the authors reduce the positivity condition to three scalar inequalities involving derivatives of $V$ with respect to the base radial coordinate $x$ and the fiber variable $w$. The most restrictive condition is:
   $$Q[V] := V_{xx} + \mu - \frac{|\mu + V_x \bar{w}|^2}{1 + V_w \bar{w}} > 0,$$
   where $\mu$ is a function of the base coordinate.

3. Potential Construction via Heat Kernel:
   The core of the construction involves creating potentials with arbitrarily large fiberwise energy while maintaining the positivity condition $Q[V] > 0$.

   • Heat Kernel Potentials: The authors utilize the heat kernel $k_t$ on the elliptic fiber $E$. They define a potential $F_\eta(t, w)$ related to the integral of the heat kernel, which possesses a fiberwise energy scaling as $2\eta^2 \log(1/t)$.
   • Global Extension and Cutoffs: A major difficulty is extending these local potentials to the compact manifold $X$. The potential must be constant for large base coordinates to ensure smoothness. The authors construct a global potential $V_\eta(x, w)$ by coupling the heat kernel time parameter $t$ to the base coordinate $x$ via a specific profile function $q(x)$.
   • Controlling Cross-Terms: The term $|\mu + V_x \bar{w}|^2$ in $Q[V]$ represents a "transport" effect that could violate positivity. The authors prove that by choosing the transport speed (determined by the derivative of $q(x)$) carefully, the negative contribution of this cross-term is dominated by the positive terms or is exponentially small. This relies on precise estimates of the heat kernel and its derivatives (Lemmas 3.2 and 3.3).
   • Correction Term: A final correction term depending only on $x$ is added to $V_\eta$ to strictly enforce $Q[V] > 0$ everywhere without altering the fiberwise energy on a fixed base ball.

Key Results
The paper proves Theorem 1.1: On the Hopf threefold $X$, there exists a smooth Hermitian form $\omega$ and a sequence of smooth functions $\phi_j$ such that $\omega + dd^c \phi_j > 0$ and
$$\int_X (\omega + dd^c \phi_j)^3 \to +\infty \quad \text{as } j \to \infty.$$
Consequently, $\text{vol}(\omega) = +\infty$, demonstrating that the Hopf threefold does not possess the bounded mass property.

Significance and Claims
The paper claims to answer a specific open question raised by Boucksom–Guedj–Lu [BGL25] regarding the bounded mass property for Hopf manifolds of dimension at least three.

• Disproof of Universality: The result disproves the hypothesis that the bounded mass property holds for every compact complex manifold. It establishes that this property fails already in complex dimension three.
• Theoretical Impact: The finiteness of the Monge–Ampère volume is a standing hypothesis in recent developments of Hermitian pluripotential theory (e.g., [ALS25; Li26]) and the theory of Bott–Chern class volumes [BGL25]. This counterexample indicates that these theories cannot be universally extended to all compact complex manifolds without additional geometric constraints.
• Methodological Contribution: The work provides a concrete mechanism for generating unbounded mass via fiberwise Dirichlet energy, leveraging the specific non-closed nature of Hermitian forms on Hopf manifolds.

The authors explicitly state that the initial counterexample was constructed with the assistance of an AI agent (Rethlas), but the simplification, rigorous proof, and all assertions in the paper are the sole responsibility of the authors.