A Matsushima theorem for K-polystable polarised smooth Fano threefolds

Abstract: We prove that if $X$ is a smooth Fano threefold and $L$ is an ample $\mathbb{Q}$-divisor such that $(X,L)$ is K-polystable, then the automorphism group $\operatorname{Aut}(X)$ is reductive. This verifies the reductivity statement predicted by the Yau--Tian--Donaldson conjecture in the setting of smooth Fano threefolds with arbitrary ample polarisation.

• The paper establishes that any K-polystable polarisation on a smooth Fano threefold forces the automorphism group to be reductive, confirming a prediction of the Yau–Tian–Donaldson conjecture.
• It employs explicit test configurations and detailed β-invariant computations across the Mori–Mukai families to demonstrate instability in cases with non-reductive automorphism groups.
• The work refines the understanding of Fano threefold moduli spaces by linking K-stability to reductivity, regardless of the choice of ample ℚ-divisor.

Matsushima-Type Theorem for K-Polystable Polarised Smooth Fano Threefolds

Introduction and Context

This work establishes that if $(X, L)$ is a K-polystable polarised smooth Fano threefold, where $L$ is an arbitrary ample $\mathbb{Q}$-divisor, then $\operatorname{Aut}(X)$ is reductive. This verifies, in the context of Fano threefolds and arbitrary ample polarisations, the reductivity property predicted by the Yau–Tian–Donaldson (YTD) conjecture. The result extends prior knowledge: previously, reductivity had been established for the case where $L = -K_X$ (the anti-canonical polarisation) and in lower-dimensional cases (del Pezzo surfaces).

K-polystability is known to be necessary for the existence of constant scalar curvature Kähler (cscK) metrics, and non-reductivity of automorphism groups is a classical obstruction (Matsushima's theorem). Thus, establishing reductivity for all K-polystable polarisations is a significant structural property of Fano threefolds. While the YTD conjecture asserts equivalence between the existence of cscK metrics and K-polystability, the "Matsushima-type" implication provides an immediate necessary condition on automorphism groups.

Overview of Methods

The proof leverages the complete classification of smooth Fano threefolds, leveraging the explicit Mori–Mukai list of 105 deformation families, and the classification of families with non-reductive automorphism groups (per Cheltsov, Shramov, Przyjalkowski). The authors systematically analyze these families, showing that all smooth Fano threefolds $X$ with $\operatorname{Aut}(X)$ non-reductive are K-unstable (or at least not K-polystable) with respect to any ample polarisation.

The destabilisation argument is made case-by-case, using three principal strategies:

• Explicit construction of test configurations with non-positive Donaldson–Futaki (DF) invariant.
• Computation of the $\beta$-invariant (valuative criterion) for divisors extracted in blow-up models, showing $\beta_L(F) < 0$ for geometrically distinguished divisors $F$, implying K-instability.
• Birational reduction, employing explicit geometric transitions to relate polarisations to simpler models where the instability may be more easily checked.

Details of the Analysis

1. Destabilisation via Test Configurations and DF Invariants

Certain threefolds with non-reductive $L$0 are shown to admit explicit test configurations with vanishing or negative DF invariant. For example, in families 2.21 and 3.13, special degenerations are constructed whose central fibers admit large, reductive automorphism groups, and for which the DF invariant vanishes. Thus, $L$1 is never K-polystable in these cases. This method, computationally guided via the Atiyah–Bott formula, also applies to families admitting torus actions with known linearizations.

2. Destabilisation via the $L$2-Invariant

For many families, the instability is detected via the $L$3-invariant (cf. Dervan–Legendre). By analyzing the birational geometry—often involving blow-ups along curves, planes, or their intersections—key divisors $L$4 (exceptional, proper transforms) are exhibited. Direct computation of $L$5, incorporating log discrepancies and volume asymptotics, shows that $L$6 for all ample polarisations $L$7, hence ruling out K-polystability.

The explicit geometric description of each threefold, using the Mori–Mukai data and further birational modifications, is critical for recognizing the relevant divisors and calculating the necessary intersection numbers.

3. Birational Reductions and Reduction to Known Cases

Some families require reducing to higher Picard rank models or particular degenerations where destabilizing divisors become apparent, or simplifying the computation of the $L$8-invariant and polarisations by passage to suitable models. The authors systematically relate the instability in complicated cases to instability in already understood cases by birational maps.

Explicit computational techniques, supported by symbolic algebra, yield closed formulas for the DF and $L$9-invariants in terms of polarisation parameters, verifying negativity in all cases of non-reductive automorphism group.

Classification and Exhaustiveness

The paper’s Table of smooth Fano threefolds with non-reductive automorphism groups corresponds to exactly the cases that must be ruled out for the main theorem. For every such family, the analysis above is carried out in detail (including those families with unique K-unstable representatives), leaving no ambiguity in the verification.

Numerical Results and Key Claims

• For each smooth Fano threefold $\mathbb{Q}$0 with non-reductive $\mathbb{Q}$1 group (from the classified families), any polarisation $\mathbb{Q}$2 yields a K-unstable or strictly K-semistable pair $\mathbb{Q}$3.
• In all cases, explicit (and in many cases, uniform) formulas for the relevant $\mathbb{Q}$4 or DF invariants are given, and their negativity is verified.
• The result is sharp for smooth structures and does not depend on the simplicity of the polarisation; arbitrary ample $\mathbb{Q}$5-divisors are allowed.

Implications and Outlook

Theoretical Implications

This work verifies, in the setting of smooth Fano threefolds and all ample polarisations, the reductivity consequence of the YTD conjecture. That is, K-polystability (and hence, conjecturally, the existence of a cscK metric) can only occur if $\mathbb{Q}$6 is reductive. The techniques further strengthen the links between algebraic K-stability theory and explicit birational geometry/classification.

The paradigm used in this paper (analyzing K-stability via $\mathbb{Q}$7-invariants and explicit test configurations in concrete birational settings) sets a model for higher dimensions, where similar classification efforts and explicit geometry may be available.

Practical and Future Directions

While reductivity is not, in general, sufficient for K-polystability (since one must also check the positivity of the DF or $\mathbb{Q}$8-invariants globally), it is a necessary filtration for moduli theory: only varieties with reductive automorphism groups can contribute stable points to K-moduli spaces. Hence, this result sharpens the construction and expected geometry of moduli spaces of polarised Fano threefolds.

Future generalizations will likely focus on singular Fano threefolds, higher-dimensional analogues, and the behavior under degenerations. As computational capacity and birational classification improve, the valuative approach exemplified here may yield more fine-grained results for general classes of varieties.

Conclusion

The paper thoroughly establishes that for all smooth Fano threefolds, the K-polystability with respect to any ample polarisation forces automorphism group reductivity, thus matching and verifying that part of the prediction of the Yau–Tian–Donaldson conjecture. The proof uses case-by-case birational and intersection-theoretic analysis, incorporating explicit test configurations and $\mathbb{Q}$9-invariant computations. This structure theorem is fundamental for both the understanding of K-stability in dimension three and for the structure of moduli spaces of Fano threefolds.