The top Yau--Yang conjecture for Kähler manifolds with positive sectional curvature

Abstract: We prove that the top wedge power of the Ricci form of a complete non-compact Kähler manifold with positive sectional curvature has finite integral. Using a result of Chen-Zhu, an immediate consequence is the quasiprojectivity of such manifolds under the assumption of bounded sectional curvature. A key new idea to prove Bézout estimates along with a Lipschitz weight with finite Monge-Ampère mass is used in the proof of the main result.

• The paper establishes that the top-degree integral of the Ricci form is finite under positive sectional curvature, advancing the Yau–Yang conjecture.
• It leverages tools like heat-flow regularization and plurisubharmonic weight construction to overcome technical challenges in non-smooth analysis.
• The results lead to a biholomorphic classification, showing that such Kähler manifolds are quasiprojective varieties with strong complex-geometric implications.

The Top Yau–Yang Conjecture for Kähler Manifolds with Positive Sectional Curvature

Introduction

The paper "The top Yau--Yang conjecture for Kähler manifolds with positive sectional curvature" (2606.19806) addresses the finiteness of the top-degree integral of the Ricci form on complete non-compact Kähler manifolds with positive sectional curvature. This problem, which generalizes the classical result of Cohn–Vossen for two-dimensional Riemannian manifolds, emerges from the Yau–Yang conjecture concerning the boundedness of curvature integrals and their geometric consequences for non-compact Kähler manifolds. The authors establish, without volume growth assumptions, that the integral $\int_X \mathrm{Ric}_\omega^n$ is finite under suitable curvature conditions. They derive significant complex-analytic and algebro-geometric implications, including biholomorphic classification results for such Kähler manifolds.

Background and Motivation

If $(M,g)$ is a complete, non-compact, two-dimensional Riemannian manifold with positive Gaussian curvature, the Cohn–Vossen theorem asserts that the total curvature is finite. In higher dimensions, Yau and then Yang formulated a conjecture for complete Kähler manifolds with non-negative holomorphic bisectional curvature, predicting that certain normalized integrals involving wedge powers of the Ricci form remain bounded. Of special interest is the case $k = n$, which asserts the global finiteness of the so-called “top Ricci mass”

$\int_X \mathrm{Ric}_\omega^n < \infty,$

which plays a central role in uniformization conjectures and in the classification and compactification of non-compact Kähler manifolds. Previous partial results had established this finiteness under additional assumptions, such as maximal volume growth or in the surface case.

Main Results

The principal claim of the paper is the following theorem:

If $(X^n, \omega)$ is a complete non-compact Kähler manifold with positive sectional curvature, then the top-degree integral of the Ricci form is finite:

$\int_X \mathrm{Ric}_\omega^n < \infty.$

The result is extended to the setting of positive holomorphic bisectional curvature and the existence of a smooth strictly plurisubharmonic exhaustion function with uniformly bounded gradient. Thus, the authors resolve a longstanding conjecture in full generality for this class of manifolds.

Moreover, as a consequence of a theorem by Chen–Zhu, the authors deduce that a manifold satisfying the above curvature conditions (with bounded sectional curvature) is biholomorphic to a quasiprojective variety, yielding strong classification and structure results in complex geometry.

Methodology and Proof Techniques

The proof leverages tools from several domains in complex and differential geometry:

• Construction of Plurisubharmonic Weights: Central to the argument is the construction of a uniformly Lipschitz strictly plurisubharmonic (psh) function $\varphi$ with $\int_X (-i\partial\bar\partial \varphi)^n < \infty$ under positive sectional curvature or positive holomorphic bisectional curvature with appropriate exhaustion.
• Monge–Ampère Mass Estimates: The authors employ Bedford–Taylor theory to handle the non-smoothness of the weight $\varphi$ and to interpret the top-degree wedge products as currents.
• Heat-Flow Regularization: To overcome the lack of smoothness, the function $\varphi$ is regularized via the heat equation using the unique positive heat kernel on stochastically complete Kähler manifolds with non-negative Ricci curvature, obtaining smooth approximations $(M,g)$0. Uniform estimates are established for these approximations.
• $(M,g)$1 Holomorphic Sections and Bochner Formulas: The existence of canonical $(M,g)$2 holomorphic sections of the canonical bundle $(M,g)$3 with respect to weighted metrics (involving the regularized psh weights) is established, using techniques akin to Hörmander’s $(M,g)$4 theory.
• Poincaré–Lelong and Bezout-Type Estimates: A key technical innovation is in handling integration-by-parts for non-smooth weights through Bedford–Taylor theory, facilitating comparison between the Ricci form and $(M,g)$5. The authors introduce a linear growth function $(M,g)$6 to aid an induction argument that supplants the delicate higher-order derivative estimates previously required.
• Cutoff Functions: Suitable cutoff functions with controlled gradients are constructed to localize integrals and facilitate passing to the limit in integration-by-parts arguments.

The proof employs an induction on the degree of terms appearing in the differential form integrals, circumventing previous limitations by exploiting the properties of the auxiliary function $(M,g)$7 and heat-flow regularization.

Notable Numerical and Structural Results

• Finiteness of Top Ricci Mass: $(M,g)$8 is established for all dimensions under the specified curvature hypotheses, without any volume growth assumptions.
• Biholomorphic Classification: Any complete non-compact Kähler manifold with positive and bounded sectional curvature is biholomorphic to a quasiprojective variety.
• Explicit Induction Scheme: The induction argument, supported by detailed estimates, yields uniform constants controlling all mixed terms in the Bedford–Taylor wedge integrals, demonstrating the robustness of their method across dimensions.
• AI Contributions: The authors remark that the application of generative AI (ChatGPT 5.5 Pro) contributed a novel functional analytic insight (specifically, the utility of $(M,g)$9 in the induction), representing a rare documented instance of original mathematical suggestion from current AI systems within geometric analysis.

Implications and Theoretical Outlook

The main theorem confirms a broad generalization of the Cohn–Vossen-type finiteness to higher-dimensional Kähler geometry under natural curvature constraints, advancing the understanding of the structure and classification of Kähler manifolds with positive curvature. The biholomorphic classification consequences connect differential-geometric curvature conditions directly to complex-algebraic structure, furthering Yau's uniformization program.

The methodological advances, particularly regarding the handling of non-smooth plurisubharmonic weights in the context of complex Monge–Ampère theory and the strategic use of heat-flow regularization, are expected to find application in related geometric analysis problems, especially those involving singular metrics or degenerate complex structures.

The explicit invocation of AI-assisted innovation in mathematical research highlights the increasing relevance of generative systems in identifying novel strategies and functional analytical tools, though, as the authors note, such AI is not yet capable of orchestrating a full proof autonomously.

Conclusion

This paper provides a definitive resolution to the top Yau–Yang conjecture in the setting of complete non-compact Kähler manifolds with positive sectional curvature, establishing the finiteness of the top Ricci mass without auxiliary hypotheses. The analytic, geometric, and algebraic methods developed yield significant classification results and enhance the technical apparatus available for future advances in Kähler geometry, complex analysis, and geometric analysis. The methodological framework developed, including functional-analytic induction schemes and innovative regularization arguments, is poised for further extension and application.