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Fujiki Class $\mathcal C$ Varieties and a Kähler Criterion

This paper establishes that flips and divisorial contractions preserve the Kähler condition for specific generalized klt pairs, provides a criterion for Kählerity within Fujiki's class $\mathcal{C}$ , and proves the existence of small $\mathbb{Q}$ -factorializations and dlt modifications for generalized pairs.

Original authors: Christopher Hacon, Yi Li, Lingyao Xie

Published 2026-08-24

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Christopher Hacon, Yi Li, Lingyao Xie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of geometry, mathematicians have long sought to understand the shapes that define our universe, from the smooth curves of a sphere to the complex, multi-dimensional structures that underpin modern physics. For decades, a powerful toolkit known as the "minimal model program" has allowed researchers to simplify these shapes, stripping away unnecessary complexity to reveal their essential forms. This program works beautifully for shapes that can be described using algebraic equations, known as projective varieties. However, a broader and more elusive category of shapes exists, called Kähler varieties. These are complex geometric spaces that possess a specific kind of smoothness and symmetry, making them central to both pure mathematics and theoretical physics. Unlike their algebraic cousins, Kähler varieties do not always behave predictably when subjected to the same simplification processes. The question of whether these shapes can be systematically reduced without losing their fundamental nature has remained a stubborn obstacle, leaving a gap in our understanding of the geometric universe.

A team of mathematicians has now taken a significant step toward bridging this gap. They have proven that a specific, critical operation used to simplify these complex shapes preserves their essential smoothness. In the process of simplification, mathematicians often encounter "kinks" or singularities that must be resolved. To do this, they perform transformations that either contract certain parts of the shape or flip them over, much like turning a page in a book. The researchers demonstrated that when these flips or contractions are performed on a specific type of complex space, the resulting shape remains a Kähler variety. This is a vital discovery because it guarantees that the entire process of simplification can continue without the shape suddenly becoming "broken" or losing the properties that make it mathematically useful. Without this assurance, the entire program of simplifying these complex spaces would risk collapsing at the first step.

The team's work also provides a new way to identify whether a complex shape belongs to this special Kähler family. They established a clear test: if a shape contains no specific type of straight line that points in a "negative" direction, then the shape is indeed Kähler. This criterion offers a practical method for mathematicians to verify the nature of these spaces without having to construct them from scratch. Furthermore, the researchers showed that even when a shape is not Kähler, it is always possible to find a closely related version that is, provided the shape meets certain mild conditions. This means that while some complex spaces may initially appear too irregular to handle, they can almost always be transformed into a well-behaved form that fits within the established mathematical framework.

The significance of these findings lies in their ability to extend the reach of the minimal model program beyond the safe harbor of algebraic geometry. By proving that the key operations of flipping and contracting preserve the Kähler condition, the authors have ensured that the program can run its full course for a much wider class of shapes. This allows mathematicians to classify and understand these complex spaces with the same confidence they have for simpler, algebraic ones. The results also clarify the boundaries of these geometric worlds, showing exactly when a shape can be simplified and when it might resist such treatment. Ultimately, this work strengthens the foundation of analytic geometry, providing the necessary tools to navigate the intricate and often hidden structures that define the mathematical universe.

Technical Summary: Fujiki Class C Varieties and a Kähler Criterion

Problem Statement
The classification of compact Kähler manifolds up to bimeromorphism is a central problem in analytic geometry. While the Minimal Model Program (MMP) has largely resolved this for projective varieties (notably via [BCHM10]), extending these results to the broader category of compact Kähler varieties remains a significant challenge. A primary obstruction is ensuring that the operations central to the MMP—specifically flips and divisorial contractions—preserve the Kähler condition. Without this preservation, the program cannot proceed within the category of compact Kähler spaces. Furthermore, there is a need for effective criteria to determine when a variety in Fujiki's class C (which includes all compact Kähler manifolds and their bimeromorphic images) is itself Kähler.

Methodology
The authors employ the framework of generalized pairs $(X, B + \beta)$ , where $\beta$ is a b-nef b-(1,1)-current. This framework allows for the treatment of transcendental problems within the MMP. The methodology relies on several technical pillars:

Local-to-Global Arguments: The proofs often proceed by establishing results locally over relatively compact Stein open subsets (where standard projective MMP results apply, e.g., [Fuj22]) and then globalizing these constructions using the properties of Fujiki's class C and rational singularities.
Generalized MMP Tools: The paper utilizes the Cone Theorem and Contraction Theorem for generalized klt pairs, as well as the existence of log canonical models.
Positivity and Cohomology: The authors work extensively with Bott–Chern cohomology $H^{1,1}_{BC}(X)$ and the generalized Mori cone $NA(X)$ . They adapt Kleiman's criterion for Kählerness, which relates the non-Kähler locus to the existence of curves trivial with respect to a nef and big class.
Resolution and Modification Techniques: The construction of small $\mathbb{Q}$ -factorializations and dlt (divisorial log terminal) modifications is central. These are achieved by running relative MMPs over log resolutions and carefully controlling discrepancies.

Key Contributions and Results

Preservation of the Kähler Condition (Theorem 1.2):
The paper proves that for a strongly $\mathbb{Q}$ -factorial compact Kähler generalized klt pair $(X, B+\beta)$ with $B+\beta_X$ big, if $f: X \to Y$ is a flipping or divisorial contraction associated to an extremal ray, then the target $Y$ is also Kähler.
- Mechanism: The proof establishes that a supporting nef and big class $\alpha$ on $X$ descends to a class $\alpha_Y$ on $Y$ . By applying a Kähler criterion (Theorem 4.3), the authors show that if $\alpha_Y$ were not Kähler, its null locus would be covered by $\alpha_Y$ -trivial rational curves. However, the contraction properties and the structure of the extremal ray prevent such curves from existing in a way that would violate the Kähler condition, thereby proving $Y$ is Kähler. This result is essential for continuing the MMP in the Kähler category.
Kähler Criterion for Fujiki Class C (Theorem 1.3 and Corollary 1.4):
The authors provide a criterion for a variety $X$ in Fujiki's class C (with generalized klt singularities) to be Kähler.
- Result: If $X$ is not Kähler, then either (1) $X$ contains a rational curve $C$ such that $-[C] \in NA(X)$ , or (2) there exists a small $\mathbb{Q}$ -factorial Kähler modification $\mu: X_{qf} \to X$ .
- Corollary: For a strongly $\mathbb{Q}$ -factorial compact variety in Fujiki's class C, $X$ is Kähler if and only if the cone $NA(X)$ does not contain a class of the form $-[C]$ for any rational curve $C$ . This generalizes a result for projective varieties from [VP21].
Existence of Modifications (Theorems 1.5 and 1.6):
- Small $\mathbb{Q}$ -factorializations: The paper proves the existence of a small projective bimeromorphic morphism $\nu: X' \to X$ such that $X'$ is normal and strongly $\mathbb{Q}$ -factorial for any compact generalized klt pair. This extends previous local results to the global setting.
- Global dlt Modifications: The authors establish the existence of a birational projective morphism $f_m: X_m \to X$ such that $X_m$ is strongly $\mathbb{Q}$ -factorial and the pair is dlt, with all exceptional divisors having discrepancy $\le -1$ . This globalizes results previously known for Stein or relatively compact settings.
Relative Minimal Model Program (Propositions 1.7 and 1.8):
The paper establishes results for the relative MMP over Moishezon contraction morphisms, showing that under specific conditions (e.g., $B+\beta_X$ modified big), the relative MMP terminates with a good minimal model or a Mori fiber space.

Significance and Claims
The paper claims to resolve a critical technical hurdle in the Kähler Minimal Model Program: the preservation of the Kähler condition under flips and divisorial contractions. This ensures that the MMP can be run entirely within the category of compact Kähler spaces, a prerequisite for proving the broader Conjecture 1.1 regarding the existence of good minimal models or Mori fiber spaces for compact Kähler varieties.

The authors explicitly state that their arguments do not rely on the recently established transcendental base point free theorem (which appears in their forthcoming work [HX26]), although they note that [HX26] relies on the results of this paper. The work is presented as a necessary step toward the full classification of compact Kähler manifolds up to bimeromorphism, analogous to the projective case. The results on small $\mathbb{Q}$ -factorializations and dlt modifications are noted as being of independent interest for the study of singularities in the analytic category.

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