K-stability of adjoint foliated structures

Abstract: We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. We also introduce a notion of Ding stability for adjoint Fano foliated structures which we show is equivalent to our notion of K-stability. We then introduce mixed alpha, beta and delta-invariants and use the reduction theorem to establish valuative criteria for the K-stability of adjoint Fano foliated structures. To conclude, as an application, we show that K-semistable adjoint Fano foliated structures with bounded volume form a bounded family.

• The paper develops a mixed K-stability theory for adjoint Fano foliated structures using F-compatible, algebraically integrable test configurations and an adjoint foliated MMP.
• It proves that stability can be tested using special degenerations, establishes equivalence with Ding stability, and gives a Fujita–Li valuative criterion through mixed beta and delta invariants.
• It shows that t-K-semistable structures with fixed dimension and a positive volume lower bound form a bounded family, while examples exhibit wall-crossing and instability as t approaches 1.

Overview

The paper develops a theory of K-stability for adjoint foliated structures, i.e. triples $(X,F,t)$ consisting of a normal variety $X$, an algebraically integrable foliation $F$, and a parameter $t\in[0,1]$, with the adjoint canonical class $K^{[t]}_{X,F}=(1-t)K_X+tK_F$. The central object is the adjoint Fano foliated structure, where $-K^{[t]}_{X,F}$ is ample. The motivation is twofold: on one hand, K-stability of Fano varieties has matured into a complete moduli theory for klt Fano varieties; on the other, the MMP for algebraically integrable adjoint foliated structures — finite generation, existence and termination of flips, and boundedness analogues of BAB — has recently been established in [CHLMSSX24, CHLMSX25]. The paper answers affirmatively whether these two threads can be combined: it defines $t$-K-stability via test configurations, proves reduction to special test configurations, establishes a Fujita–Li-type valuative criterion via prime divisors, introduces mixed $\alpha$, $\beta$, $\delta$-invariants, and derives boundedness of $X$0-K-semistable structures.

F-compatible test configurations and the mixed Donaldson–Futaki invariant

The key definitional innovation is the notion of an $X$1-compatible test configuration: a normal test configuration $X$2 for $X$3 equipped with a $X$4-equivariant saturated integrable subsheaf $X$5 restricting to the product foliation over the punctured base, with $X$6 required to be algebraically integrable. The author is explicit that this last condition is imposed in order to invoke the MMP results of [CHLMSSX24, CHLMSX25], and that it is not claimed to be preserved under arbitrary deformations — removing it is left as an open question.

For such a test configuration, the mixed Donaldson–Futaki invariant is defined intersection-theoretically:

$X$7

where $X$8 is the foliated slope. The definition passes natural sanity checks: at $X$9 or when $F$0 (or $F$1) it recovers the classical Donaldson–Futaki invariant. Positivity against all normal $F$2-compatible test configurations defines $F$3-K-semistability, with strict positivity for non-trivial configurations giving $F$4-K-stability, and uniform stability formulated via the non-Archimedean J-functional. At $F$5 one obtains a purely foliated notion, but the paper shows (Remark 6.x) that a naive K-stability theory for Fano foliations alone would be vacuous: any non-trivial $F$6-invariant divisor $F$7 satisfies $F$8 while $F$9, forcing $t\in[0,1]$0. This observation justifies focusing on the genuinely mixed regime $t\in[0,1]$1.

Reduction to special test configurations

The technical core adapts the Li–Xu road map [LX14] to families of adjoint Fano foliated structures over a curve. Starting from a normal $t\in[0,1]$2-compatible test configuration, after a finite base change and equivariant semistable resolution, the paper constructs a $t\in[0,1]$3-factorial qdlt model using the qdlt modification theorem for adjoint foliated structures, then runs a $t\in[0,1]$4-equivariant $t\in[0,1]$5-MMP with scaling, reaching a relative anti-adjoint model $t\in[0,1]$6. A final extraction step, ordering vertical coefficients so that exactly one distinguished component survives, produces a special fibre of discrepancy zero and hence a normal $t\in[0,1]$7-compatible special test configuration.

The monotonicity mechanism is that the mixed Donaldson–Futaki invariant is non-increasing along each birational step: along the qdlt/base-change step via differentiation of the invariant under perturbation of the polarisation (using the negativity of intersections of fibre-supported divisors), along divisorial contractions and flips because in the anti-adjoint regime $t\in[0,1]$8 and $t\in[0,1]$9 increases under the MMP steps, and along the final contraction by an interpolation argument. The main result is that for any normal $K^{[t]}_{X,F}=(1-t)K_X+tK_F$0-compatible test configuration there exists, after a base change of degree $K^{[t]}_{X,F}=(1-t)K_X+tK_F$1, a special $K^{[t]}_{X,F}=(1-t)K_X+tK_F$2-compatible test configuration with

$K^{[t]}_{X,F}=(1-t)K_X+tK_F$3

Consequently, $K^{[t]}_{X,F}=(1-t)K_X+tK_F$4-K-semistability can be tested only against special $K^{[t]}_{X,F}=(1-t)K_X+tK_F$5-compatible test configurations. Two assumptions are load-bearing here: the ambient variety must be potentially klt and $K^{[t]}_{X,F}=(1-t)K_X+tK_F$6 not pseudo-effective over the base, both needed to run the adjoint foliated MMP; the author notes that future work will show $K^{[t]}_{X,F}=(1-t)K_X+tK_F$7-K-semistable structures are klt, partially justifying these hypotheses.

Ding stability and the valuative criterion

A mixed Ding invariant is defined via the mixed log canonical threshold of the mixed correction divisor $K^{[t]}_{X,F}=(1-t)K_X+tK_F$8 along the central fibre. The same birational procedure shows the quantity $K^{[t]}_{X,F}=(1-t)K_X+tK_F$9 decreases along each step, and since the resulting special configuration is $-K^{[t]}_{X,F}$0-weakly special (on which $-K^{[t]}_{X,F}$1), the paper obtains the equivalence of $-K^{[t]}_{X,F}$2-K-semistability with $-K^{[t]}_{X,F}$3-Ding semistability, and of uniform $-K^{[t]}_{X,F}$4-K-stability with uniform $-K^{[t]}_{X,F}$5-Ding stability.

Combining this with a foliated adaptation of Fujita's flag-ideal blow-up construction yields the Fujita–Li valuative criterion: for $-K^{[t]}_{X,F}$6 and $-K^{[t]}_{X,F}$7,

$-K^{[t]}_{X,F}$8

for every prime divisor $-K^{[t]}_{X,F}$9 over $t$0, where $t$1. A key computational input is Proposition 4.x showing that for a Rees degeneration induced by an $t$2-dreamy valuation, $t$3; the proof carefully tracks how the foliated discrepancy transforms under blowing up $t$4, contributing exactly $t$5 (the transversality indicator) in the transverse case. It follows that $t$6 of the associated special test configuration equals the mixed $t$7-invariant of the central-fibre valuation.

Mixed $t$8, $t$9-invariants and boundedness

Following Blum–Jonsson [BJ20], the paper defines the mixed stability threshold via basis type divisors,

$\alpha$0

and proves the limit of the finite-level invariants exists and admits this valuation-theoretic description. Consequently, $\alpha$1 is $\alpha$2-K-semistable if and only if $\alpha$3, and uniformly $\alpha$4-K-stable if and only if $\alpha$5. The mixed $\alpha$6-invariant $\alpha$7 satisfies the standard sandwich

$\alpha$8

so $\alpha$9 suffices for $\beta$0-K-semistability, while $\beta$1-K-semistability implies $\beta$2.

This lower bound feeds directly into boundedness. Using weighted blow-up computations showing that mixed discrepancies of extracted divisors satisfy $\beta$3, the paper proves that any family with $\beta$4 and $\beta$5 consists of $\beta$6-lc structures with $\beta$7; combined with the boundedness theorem for adjoint Fano foliated structures from [CHLMSX25, Theorem B], this yields that $\beta$8-dimensional $\beta$9-K-semistable adjoint Fano foliated structures with volume bounded below form a bounded family. The author notes that a related boundedness statement appears in [CLSV26, Theorem E], proved there via minimal log discrepancy estimates without an accompanying K-stability notion; that result can also be deduced from the present theorem.

Examples and wall-crossing behaviour

Several general criteria illustrate the theory. If $\delta$0 is K-semistable $\delta$1-Fano, $\delta$2 is lc, and $\delta$3 with $\delta$4 for all divisorial valuations, then $\delta$5 is $\delta$6-K-semistable whenever $\delta$7; e.g., a general pencil of plane cubics gives a $\delta$8-K-semistable structure on $\delta$9 for all $X$00. If $X$01 is uniformly K-stable and $X$02 is lc, then $X$03 is uniformly $X$04-K-stable for all sufficiently small $X$05, by continuity of the delta invariant on the big cone.

On the unstable side, the paper proves a structural dichotomy: for any algebraically integrable Fano foliation $X$06, there exists $X$07 such that $X$08 fails $X$09-K-semistability for all $X$10, since an $X$11-invariant divisor has $X$12 while $X$13 stays positive near $X$14. Similarly, non-lc singularities of $X$15 obstruct semistability near $X$16, so semistability along a sequence $X$17 forces $X$18 to be lc. In the proportional case $X$19, the affine-linearity of $X$20 in $X$21 implies the semistable locus $X$22 is a closed interval, possibly empty — a wall-crossing phenomenon. A concrete example: for a smooth cubic fourfold $X$23 with the codimension-one foliation induced by a general Lefschetz pencil, $X$24 is uniformly $X$25-K-stable for small $X$26 but not $X$27-K-semistable for $X$28, since the smooth pencil member $X$29 has $X$30. The radial foliation on $X$31 gives instability for every $X$32.

Limitations and open questions

Several restrictions are acknowledged explicitly. The algebraic integrability requirement on $X$33 in the definition of $X$34-compatible test configurations is a working hypothesis rather than a canonical feature, and its removability is open. The reduction and Ding arguments require the ambient variety to be potentially klt and the relative adjoint class to be non-pseudo-effective; the justification that $X$35-K-semistable structures satisfy these conditions is deferred to future work. The openness of the $X$36-K-semistable locus — the other essential ingredient for constructing a moduli stack $X$37 with a good moduli space $X$38 of $X$39-K-polystable objects — "seems out of reach with the current MMP methods" according to the author. Finally, the toolkit for verifying $X$40-K-stability in explicit examples remains limited, motivating the question of whether Abban–Zhuang theory extends to the mixed setting; the paper also leaves the properties of the mixed normalised volume $X$41 unexplored, with a cone-characterisation of $X$42-K-semistability planned for subsequent work.

Conclusion

The paper establishes that the standard architecture of K-stability for Fano varieties — test configurations, special reduction via MMP, Ding equivalence, Fujita–Li valuative criteria, delta-invariant characterisations, and boundedness — carries over coherently to adjoint Fano foliated structures, provided one works with $X$43-compatible, algebraically integrable degenerations and exploits the recently developed adjoint foliated MMP. The boundedness theorem for $X$44-K-semistable structures supplies the first pillar of a potential moduli theory, whose completion hinges on openness of the semistable locus and on $X$45-completeness/$X$46-reductivity checks formulated through test configurations.