An approach to the abundance conjecture for Kähler varieties via algebraic reduction

Abstract: In this article, we establish a strategy to the abundance conjecture for Kähler varieties via induction on algebraic dimension. Our strategy is to reduce the abundance conjecture for Kähler varieties to the abundance conjecture for projective varieties using the algebraic reduction fibration. In dimension 4, we apply our inductive strategy to obtain some cases of the abundance conjecture for Kähler fourfolds that are not algebraic or have trivial $K_X$.

• The paper establishes an inductive strategy linking abundance on Kähler varieties to projective cases via algebraic reduction.
• It refines Zariski decomposition and the canonical bundle formula for analytic settings using Hodge-theoretic local positivity methods.
• The work proves semi-ampleness for Kähler fourfolds with algebraic dimension three, unifying cases in the analytic minimal model program.

Inductive Approach to the Abundance Conjecture for Kähler Varieties via Algebraic Reduction

Introduction and Context

The abundance conjecture in birational geometry asserts that for a log canonical (lc) pair $(X, \Delta)$ with $K_X + \Delta$ nef, the adjoint divisor $K_X + \Delta$ should be semiample. While this conjecture is resolved for projective varieties in dimensions up to three and for Kähler surfaces, its general validity for higher-dimensional compact Kähler varieties remains open. The work "An approach to the abundance conjecture for Kähler varieties via algebraic reduction" (2604.03879) advances the study of Kähler varieties by introducing a novel inductive paradigm that leverages the algebraic reduction map, thereby linking the conjecture for Kähler varieties to the well-understood projective case.

Algebraic Reduction and Inductive Schemes

The algebraic reduction of a compact analytic variety $X$ yields a projective variety $Y$ and a dominant meromorphic fibration such that the function field $\mathbb{C}(X) = \mathbb{C}(Y)$, with the dimension $a(X) = \mathrm{tr.deg}_{\mathbb{C}} \mathbb{C}(X)$ termed the algebraic dimension. The algebraic reduction induces a filtration of the abundance problem, suggesting an inductive split: analyzing semi-ampleness on the base $Y$, typically projective, and the generic fiber, a Kähler variety of strictly lower dimension.

The main theorem formalizes this: if the minimal model program (MMP) and abundance hold for projective varieties in dimension $a(X)$, and abundance holds for Kähler varieties of dimension $\dim(X) - a(X)$, then, under mild technical assumptions (notably, the almost holomorphicity of algebraic reduction and properties of the moduli b-divisor), abundance holds for $K_X + \Delta$0. This provides a robust inductive scaffold for higher dimensions, contingent on the structure of the algebraic reduction.

Key Technical Ingredients

Zariski Decomposition in Kähler Geometry

The work refines the notion of Zariski decomposition for Kähler varieties, adapting b-divisor and nef-part compatibility arguments from the projective to the analytic setting. Notably, it introduces Hodge-theoretic local positivity methods to replace reliance on hyperplane sections, which may be absent in general Kähler varieties.

A main technical result is the compatibility of Zariski decomposition under pullbacks by morphisms from Kähler varieties to projective bases, crucial for reducing nefness and semi-ampleness questions to lower-dimensional or more algebraic settings.

Canonical Bundle Formula and Moduli Part Positivity

The canonical bundle formula, traditionally originating in the context of fibered algebraic varieties, is established for morphisms from Kähler varieties to projective varieties by adapting analytic and semi-positivity techniques. The moduli b-divisor $K_X + \Delta$1 arising in the canonical bundle formula encodes the variation of the canonical class in the family; its positivity (b-nef, b-goodness) is essential for inductive arguments. The paper extends projective techniques of Ambro, Kawamata, and others, proving that in the analytic-proper setting with projective base, the necessary positivity of $K_X + \Delta$2 holds.

Analytic Semistable Reduction

Semi-stable reduction in codimension one, enabling normal crossings and toroidal structures over codimension-one loci after suitable base changes, is extended to complex analytic Kähler settings. Considerable attention is paid to analytic subtleties regarding ideal sheaf extensions and normality, ensuring that the reduction arguments from the algebraic case remain valid.

New Results for Kähler Fourfolds

As a concrete application, the techniques solve new instances of the abundance conjecture for Kähler fourfolds. Specifically, if $K_X + \Delta$3 is a compact Kähler fourfold with algebraic dimension $K_X + \Delta$4 (i.e., algebraic reduction yields a 3-fold base), and $K_X + \Delta$5 is nef, the paper establishes that $K_X + \Delta$6 is semiample—a result not previously available for non-algebraic Kähler fourfolds. Similarly, the results for Kähler threefolds with $K_X + \Delta$7 are recovered and unified, detailing how the b-divisor hypotheses and almost-holomorphicity are satisfied in those cases.

Implications and Future Perspectives

The proposed strategy demonstrates the structural depth provided by the algebraic reduction in analytic geometry, allowing for a systematic reduction of the abundance conjecture for Kähler varieties to projective and lower-dimensional Kähler settings. The compatibility of various decomposition, bundle, and positivity results in the analytic category is central, marking a matured synthesis of Hodge theory, analytic MMP, and the advanced use of b-divisors.

If the technical assumptions—b-goodness of the moduli part, almost holomorphicity of algebraic reduction—can be established generally (as suggested, for instance, by ongoing progress in the study of the moduli b-divisor for klt-trivial fibrations of relative dimension one), the inductive program outlined could yield a resolution of the abundance conjecture for broad classes of compact Kähler varieties. The links with the invariance of plurigenera further enhance the theoretical significance.

Future progress will likely revolve around refining the analytic techniques for semi-stable reduction, extending the b-divisor theory in the analytic and Kähler context, and establishing the necessary positivity results in general. The analytic tools introduced are also likely to find further applications in the minimal model program for complex analytic spaces.

Conclusion

This work provides a rigorous and highly technical framework for addressing the abundance conjecture in the Kähler category by connecting it with the more classical projective setting via algebraic reduction. By constructing an inductive machinery that integrates Zariski decomposition, the canonical bundle formula, Hodge theory, and analytic MMP techniques, the paper achieves new cases for Kähler fourfolds and establishes a methodological template for future advances (2604.03879).