Failure of the semi log canonical Abundance for compact Kähler threefolds

Abstract: In this article we show that the semi log canonical abundance for compact Kähler varieties fails in dimension $3$. More specifically we construct a counterexample of a compact Kähler (irreducible) slc threefold $(X, 0)$ such that $K_X$ is nef and $κ(\tilde X, K_{\tilde X}+\tilde D)=0$, where $μ:(\tilde X, \tilde D)\to X$ is the normalization morphism, but $K_X$ is not semiample. On the other hand, we show that if we start with a compact Kähler semi-dlt pair, then the abundance does hold, i.e., if $(X, Δ)$ is a compact Kähler sdlt pair of dimension $3$ such that $K_X+Δ$ is nef, then it is semiample. We also show that if $(X, Δ)$ is a compact Kähler slc pair of dimension $3$, $K_X+Δ$ is nef, and $κ(X'_i, Δ'_i+D'_i)>0$ for all $i$, where $μ:\sqcup(X'_i, Δ'_i+D'_i)\to (X,Δ)$ is the normalization, then $K_X+Δ$ is semiample.

• The paper constructs a counterexample demonstrating that a compact Kähler slc threefold with a nef canonical divisor can fail to be semiample.
• It establishes that for sdlt pairs, the canonical divisor is semiample by adapting Fujino's projective techniques to the analytic setting.
• It further shows that slc threefolds with positive log Kodaira dimension in all normalization components achieve abundance through minimal admissibility of pluricanonical sections.

Failure of the Semi Log Canonical Abundance for Compact Kähler Threefolds

Introduction and Context

The abundance conjecture, central in the Minimal Model Program (MMP), posits that for a variety $X$ with certain singularities and a nef canonical divisor (or log canonical divisor $K_X + \Delta$ for a pair), this divisor should actually be semiample—that is, some multiple is globally generated and defines a morphism. In dimension three, this statement has been a cornerstone in both algebraic and complex-analytic (Kähler) geometry. The abundance theorem for projective threefolds and for compact Kähler threefolds with log canonical (lc) singularities has been established. The present work addresses whether abundance holds for the broader class of compact Kähler threefolds with semi log canonical (slc) singularities.

Main Results

This paper delivers two principal structural results alongside an explicit counterexample to abundance in the slc setting:

1. Counterexample to slc Abundance: The paper constructs a compact Kähler slc threefold $(X,0)$ with nef canonical divisor such that $K_X$ is not semiample, and in fact $H^0(X, mK_X) = 0$ for all large $m$—a clear failure of the abundance principle. This construction exploits the failure of finiteness of the pluricanonical representation on certain non-algebraic K3 surfaces and leverages the gluing formalism for analytic spaces [(2604.28085), Section: slc-abundance].
2. Abundance for sdlt Kähler Threefolds: For the subclass of semi divisorial log terminal (sdlt) pairs in dimension three, if $(X, \Delta)$ is compact Kähler and $K_X+\Delta$ is nef, then $K_X+\Delta$ is semiample. The proof involves an analytic adaptation of Fujino's projective techniques and crucially circumnavigates the breakdown of finite pluricanonical representation by restricting to the sdlt context.
3. Conditional Abundance for slc Pairs with Positive Log Kodaira Dimension: If $(X, \Delta)$ is a compact Kähler slc threefold with $K_X + \Delta$0 nef, and the normalization $K_X + \Delta$1 satisfies positive log Kodaira dimension for all components, then $K_X + \Delta$2 is also semiample.

Methodological Overview

The Projective Case and Failure in the Analytic Category

In the projective setting, abundance for slc threefolds follows from Fujino's approach, which deeply relies on the finiteness of the $K_X + \Delta$3-pluricanonical representation for log canonical surfaces, itself resting on classical theorems for Moishezon manifolds. However, this finiteness fails for non-algebraic compact Kähler surfaces, particularly certain K3 surfaces admitting automorphisms of infinite order that act non-trivially on pluricanonical forms.

The Counterexample Construction

The counterexample exploits this failure:

• Let $K_X + \Delta$4 be a Kähler, non-algebraic K3 surface with a non-symplectic automorphism of infinite order.
• Let $K_X + \Delta$5 with conductor divisor $K_X + \Delta$6 and glue $K_X + \Delta$7 to $K_X + \Delta$8 via the automorphism.
• The resulting $K_X + \Delta$9 is slc, $(X,0)$0 is nef, but every pluricanonical section on the normalization fails the necessary Galois invariance to descend to $(X,0)$1, hence $(X,0)$2 is not semiample.

Abundance for sdlt and Positive Log Kodaira Dimension Cases

To restore abundance in special cases, the paper:

• Develops analytic analogues of projective techniques through minimally admissible and minimally preadmissible sections, which strengthen standard admissibility notions to guarantee descent and generation even if plenary $(X,0)$3-pluricanonical finiteness fails globally.
• Proves that for compact Kähler sdlt threefolds (where normalization components are normal, and the non-normality is mild), minimal admissibility suffices to ensure semiampleness.
• Demonstrates via careful descent properties and an analytic version of the connectedness principle that abundance also holds for slc pairs whose normalization components all have positive log Kodaira dimension.

Rigorous Implications

• The abundance theorem for compact Kähler threefolds fails in general for slc pairs. The mechanism is rooted in transcendental features absent in the projective case.
• For semi divisorial log terminal (sdlt) pairs, and for slc pairs whose normalizations have positive log Kodaira dimension, abundance is still valid. This identifies the obstructions to abundance as genuinely analytic in nature, tied to the possible non-finiteness of pluricanonical representations in dimension two.
• The paper provides explicitly constructible counterexamples in all dimensions $(X,0)$4 via dimension-raising constructions (products with an elliptic curve), clarifying the geometric origin of the anomaly [(2604.28085), Section: slc-abundance].

Numerical and Structural Highlights

• The counterexample yields $(X,0)$5 for all large $(X,0)$6.
• In contrast, for sdlt compact Kähler threefolds $(X,0)$7 with nef $(X,0)$8, there exists a sufficiently large $(X,0)$9 such that $K_X$0 is globally generated.

Theoretical and Practical Implications

• The results showcase the necessity of projective and semi-ampleness conditions in abundance conjectures, emphasizing that analytic-geometric phenomena (like infinite order automorphisms of K3s) can defeat algebraic intuitions.
• For the birational classification program in Kähler geometry, the precise delineation of when abundance holds informs both the formulation of conjectures and the design of higher-dimensional MMP strategies beyond the algebraic case.
• Practically, the construction and descent formalism for sections over slc varieties have implications for the study of analytic moduli, degenerations, and the topology of non-algebraic Kähler spaces.

Potential for Future Developments

• The analytic techniques for constructing and descending pluricanonical sections may influence the study of moduli of Kähler varieties.
• Extensions to further general singularities or other analytic settings (e.g., Kähler spaces with additional structures such as special holonomy) are plausible lines of future research.
• The explicit examples clarify the necessary conditions for effective MMP-type results in transcendental geometry and may inform advances in the study of Kähler-Einstein metrics on singular spaces and their moduli.

Conclusion

This work establishes a sharp dichotomy in the abundance problem for compact Kähler threefolds: failure occurs in full generality for slc varieties, rooted in analytic phenomena absent from the projective setting, while abundance holds under additional structure (sdlt, positive log Kodaira dimension). The techniques are precise, the boundary between algebraic and analytic categorically marked, and the implications for higher-dimensional birational geometry are both definitive and generative for subsequent research (2604.28085).