复变函数：2025–2026 年度综述

An annual survey of the math.CV arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 108 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Bergman Spaces and Hyperbolic Geometry in Complex Domains (85.0%) — Focuses on Bergman projections, hyperbolic metrics, and geometric function theory in planar and higher-dimensional complex domains.
• Holomorphic Functions on Complex Manifolds and Kähler Geometry (7.1%) — Investigates rigidity, Monge-Ampère flows, and cohomology on compact Kähler and non-Kähler complex manifolds.
• Geometric Function Theory and Univalent Function Estimates (4.7%) — Addresses coefficient bounds, Landau-type theorems, and geometric properties of analytic and harmonic univalent functions.
• Operator Theory in Hardy and Fock Spaces (1.3%) — Analyzes Toeplitz operators, Hardy spaces, and Fock space structures within operator algebras.
• Singularities and Local Solutions in Complex Differential Equations (0.6%) — Explores existence, extension, and local behavior of solutions to differential and difference equations near singular points.
• Polynomial Dynamics and Zero Distribution (0.6%) — Studies rigidity, zero sets, and combinatorial properties of polynomials and their associated dynamical systems.
• Conjectures and Counterexamples in Complex Analysis (0.3%) — Provides proofs, disproofs, and precise constructions related to major conjectures like Jacobian and Agler-McCarthy.
• Generalized Function Theory and Dimensional Structures (0.2%) — Extends function theory to complexified octonions and explores dimensional aspects of Bergman spaces and Galois theory.
• Kähler-Einstein Metrics and Metric Geometry Rigidity (0.2%) — Examines rigidity, curvature, and metric geometry in the space of Kähler metrics, including recent conjecture disproofs.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.