Complex Monge–Ampère Pullback in Complex Geometry

• Complex Monge–Ampère pullback is a method to extend the degenerate operator on complex manifolds, enabling treatment of singular and non-invertible maps.
• It employs precise cohomological and measure-theoretic conditions to prevent spurious mass concentrations and ensure well-defined wedge products.
• Applications span complex dynamics, intersection theory, and degenerate Monge–Ampère equations, offering robust tools for analyzing singular geometric flows.

A complex Monge–Ampère pullback is a construction in pluripotential theory and complex geometry that allows well-defined extension of the (degenerate) complex Monge–Ampère operator under holomorphic maps, including those with mild indeterminacy. This is essential in complex dynamics, complex Monge–Ampère equations, and intersection theory on complex manifolds, particularly in the setting of currents and non-Kähler or singular geometry. The theory addresses the analytic and measure-theoretic subtleties present when pulling back Monge–Ampère-type quantities along generically non-invertible or singular maps.

1. Framework for the Monge–Ampère Pullback

Let $X$ and $Y$ be compact complex manifolds. The core objects of interest are currents of type $(1,1)$ on $Y$, typically written in a decomposition $T = T^+ - T^-$ where $T^\pm$ are positive closed $(1,1)$-currents, each smooth outside finitely many points (Class (A)). Fundamental to the pullback construction is a holomorphic or meromorphic map $f:X \to Y$. For compact Kähler threefolds, the most studied case is a pseudo-isomorphism: a bimeromorphic map whose restriction to the complements of the respective indeterminacy loci, $f : X \setminus I(f) \to Y \setminus I(f^{-1})$, is an isomorphism, with both $I(f)$ and $Y$0 analytic subsets of complex dimension at most one. This enables extending the pullback of currents in a controlled manner beyond the domain where $Y$1 is holomorphic (Truong, 2014).

When considering pullbacks of forms via surjective holomorphic maps between compact Hermitian (not necessarily Kähler) manifolds, the pullback structure $Y$2 (with $Y$3 a Hermitian form on $Y$4) yields degenerate, potentially non-closed, big $Y$5-forms on $Y$6 (Dang, 26 Mar 2026).

2. Conditions for Well-definedness

The measure-theoretic and topological difficulties of extending the Monge–Ampère operator under pullback through singular or bimeromorphic maps necessitate three critical conditions when $Y$7:

• Condition 1 (Cohomological pairing): For every irreducible curve $Y$8, the intersection $Y$9.
• Condition 2 (Absence of Dirac mass for cubic wedge): The signed measure $(1,1)$0 has no mass supported on $(1,1)$1.
• Condition 3 (Absence of Dirac mass along curves): For every irreducible curve $(1,1)$2, the measure $(1,1)$3 has no point masses.

For positive smooth or positive Class (A) currents, these conditions automatically hold or reduce to cohomological requirements (Truong, 2014). The necessity of these conditions is rooted in avoiding concentration of Monge–Ampère mass on the analytic subset where pullback is subtle or undefined, ensuring the operator does not pick up unwanted singularities.

3. Definition and Monotone Approximation

Once Conditions 1–3 are satisfied, there exists a uniquely defined closed $(1,1)$4-current $(1,1)$5 on $(1,1)$6, with cohomology class $(1,1)$7, expressible globally as $(1,1)$8 where $(1,1)$9 is a smooth closed $Y$0-form and $Y$1 is a difference of quasi-psh functions. To define the associated Monge–Ampère operator $Y$2, Bedford–Taylor's theory is used:

• Push-forward construction: For smooth test functions $Y$3, the operator is specified by integrating against the push-forward $Y$4, defined via monotone convergence using approximations by smooth quasi-psh functions.
• Wedge product: The three-fold Monge–Ampère wedge $Y$5 is made precise through this push-forward procedure, ensuring independence from the choice of decomposition or potential representatives (Truong, 2014).

Bedford–Taylor monotone convergence ensures the stability of the operator under smooth approximations ($Y$6), as well as under Demailly “good” approximations of $Y$7 by smooth currents (Truong, 2014).

4. Regularity, Continuity, and Singularities

If $Y$8 is smooth or positive in Class (A) and conditions are satisfied, the Monge–Ampère measure coincides with the naive pullback: $Y$9 (Truong, 2014). However, the pullback current and associated Monge–Ampère measure can be singular along the indeterminacy loci; even if $T = T^+ - T^-$0 is smooth, the iterated pullback may introduce pole-type singularities.

On compact Hermitian manifolds and for degenerate complex Monge–Ampère equations with pullback reference forms, it has been shown that the Monge–Ampère potential is globally continuous under the pullback structure ($T = T^+ - T^-$1), extending continuity and Hölder regularity results to non-Kähler settings (Dang, 26 Mar 2026). Hölder continuity of the solution and quantitative stability follow via Demailly’s smoothing and capacity estimates on the ample locus of $T = T^+ - T^-$2.

5. Cohomological and Measure-theoretic Mechanisms

The pullback construction leverages both cohomological and measure-theoretic arguments:

• Cohomologically, Condition 1 ensures that no residual mass is assigned along curves collapsed by $T = T^+ - T^-$3.
• Measure-theoretically, Condition 2 guarantees that wedge products of $T = T^+ - T^-$4 with itself and with currents of integration do not create spurious Dirac masses on indeterminacy loci. Bedford–Taylor products and push-forwards rely on classical regularization and monotone convergence schemes.

The construction is stable under approximation; when a sequence of smooth approximating currents converges, their Monge–Ampère measures converge weakly to that of the limit, ensuring analytic compatibility for applications in complex geometry (Truong, 2014).

6. Applications and Geometric Outcomes

The complex Monge–Ampère pullback is fundamental in several contexts:

• Complex dynamics and intersection theory: The pullback allows investigation of invariant measures or dynamical degrees for complex dynamics on threefolds and links intersection-theoretic behavior under birational or generically non-invertible maps.
• Monge–Ampère equations on degenerate backgrounds: Regularity results for potentials solving degenerate Monge–Ampère equations in big classes produced via pullback structures ($T = T^+ - T^-$5), inform global continuity, sharp Hölder estimates on ample loci, and provide stability bounds (Dang, 26 Mar 2026).
• Uniform geometric bounds: The structure enables derivation of uniform diameter bounds under geometric flows (twisted Chern–Ricci flow) when the reference form is a pullback, establishing metric uniformity in evolving Hermitian settings.

Setting	Pullback Form	Regularity Results
Kähler threefold	Pseudo-isomorphism	Well-defined, continuous Monge–Ampère
Hermitian manifold	Pullback of metric	Global continuity, Hölder estimates

A plausible implication is the expanding relevance of these methods for parabolic flows and non-Kähler geometry, where classical Monge–Ampère theory is insufficient.

7. Consequences and Scope

The development of the complex Monge–Ampère pullback under pseudo-isomorphisms and in degenerate settings substantially enlarges the analytic toolkit for studying complex structures with singularities or non-invertible morphisms. By guaranteeing regularity and well-definedness under natural geometric assumptions, these techniques underpin advanced results in logarithmic birational geometry, pluripotential theory, and the metric study of complex spaces (Truong, 2014, Dang, 26 Mar 2026).