Log Canonical Foliated Pairs

• Log canonical foliated pairs are defined on normal varieties with a foliation and boundary where discrepancy inequalities account for invariant and non-invariant divisors.
• They enable explicit treatments in surfaces and threefolds, using resolution graphs, minimal model programs, and separatrix arc-space analysis to handle singularity and tangency issues.
• Their framework bridges toric and birational geometry, yielding ACC results, boundedness of minimal discrepancies, and novel adjunction formulas applicable in both classical and foliated settings.

Log canonical foliated pairs are the foliated analogue of log canonical pairs in birational geometry. A foliated pair consists of a normal variety $X$, a foliation $\mathcal F\subset T_X$ that is saturated and closed under Lie bracket, and a boundary $\Delta$ such that $K_{\mathcal F}+\Delta$ is $\mathbb Q$-Cartier; log canonicity is then defined by a discrepancy inequality for divisors over $X$, with a correction that depends on whether the divisor is invariant under the induced foliation. In recent work, this framework has acquired several distinct but compatible incarnations: a surface theory with explicit resolution graphs and minimal log discrepancies, a threefold minimal model program for corank-one and rank-one foliations, and a tangential arc-space theory that isolates the separatrix sector and reduces discrepancy calculations to ordinary log pairs on normalized branches and conductors (Chen, 2021, Chaudhuri et al., 2024, Corrêa, 2 Jul 2026).

1. Discrepancies, invariance, and basic definitions

A foliation on a normal variety is a coherent saturated subsheaf of the tangent sheaf that is closed under the Lie bracket. On surfaces one usually works with rank-one foliations, while on threefolds an important case is co-rank one foliations, locally given on the smooth locus by an integrable $1$-form unique up to a unit. The canonical divisor $K_{\mathcal F}$ is defined from the determinant of the foliation: on surfaces $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$, and for a corank-one foliation on a normal projective threefold it is the Weil divisor class associated to the determinant of the conormal sheaf of $\mathcal F$ (Chen, 2021, Chaudhuri et al., 2024).

For a proper birational morphism $\mathcal F\subset T_X$0 with induced foliation $\mathcal F\subset T_X$1, the discrepancy of a divisor $\mathcal F\subset T_X$2 over $\mathcal F\subset T_X$3 is defined by comparing $\mathcal F\subset T_X$4 with the pullback of $\mathcal F\subset T_X$5. In the McQuillan convention adopted in the surface and corank-one threefold literature, one sets

$\mathcal F\subset T_X$6

and defines

$\mathcal F\subset T_X$7

The pair is log canonical if $\mathcal F\subset T_X$8 for every divisor $\mathcal F\subset T_X$9 over $\Delta$0, and klt if the inequality is strict. Canonical and terminal singularities are defined by the usual inequalities $\Delta$1 and $\Delta$2 (Chen, 2021, Chaudhuri et al., 2024, Chen, 2022).

This invariant/transverse dichotomy is the first major difference from ordinary log pairs. Invariant divisors satisfy the stricter bound $\Delta$3, whereas non-invariant divisors satisfy the classical-looking bound $\Delta$4. A recurring theme across the subject is that many phenomena depend on whether a divisor or curve is tangent to the foliation, not merely on its ambient birational behavior (Chen, 2022).

A variant convention appears in the rank-one threefold base-point-free theory: there the discrepancy formula is written with the strict transform of $\Delta$5 together with all non-$\Delta$6-invariant exceptional divisors added to $\Delta$7 with coefficient $\Delta$8, and the lc condition is then stated as $\Delta$9. In that setup a basic consequence is that, if $K_{\mathcal F}+\Delta$0 is log canonical, no irreducible component of $K_{\mathcal F}+\Delta$1 is $K_{\mathcal F}+\Delta$2-invariant (Cascini et al., 3 Sep 2025).

2. Surface singularities and local structure

On surfaces, log canonical foliated pairs admit a highly explicit local theory. If $K_{\mathcal F}+\Delta$3 is a non-invariant reduced curve on a smooth surface, the tangency order satisfies

$K_{\mathcal F}+\Delta$4

If $K_{\mathcal F}+\Delta$5 is invariant, then one has the index identities

$K_{\mathcal F}+\Delta$6

where $K_{\mathcal F}+\Delta$7 is the local index along invariant curves and $K_{\mathcal F}+\Delta$8 is the Camacho–Sad index. At a non-degenerate singularity with local model $K_{\mathcal F}+\Delta$9, the two invariant branches satisfy $\mathbb Q$0 and the Camacho–Sad residues are $\mathbb Q$1 and $\mathbb Q$2; at a saddle-node, the strong separatrix has $\mathbb Q$3 and $\mathbb Q$4, while the weak separatrix, when present, carries the higher $\mathbb Q$5-index $\mathbb Q$6 (Chen, 2021).

The minimal foliated resolution is governed by Seidenberg reduction, but it is not identical to the minimal resolution of the underlying surface singularity. For a germ $\mathbb Q$7 with log canonical foliation singularity, the weighted dual graph of the exceptional divisor on the minimal resolution belongs to one of seven types. These include an $\mathbb Q$8-chain; a chain of three invariant curves with two $\mathbb Q$9-$X$0-curves and a bad tail; a chain of $X$1-$X$2-curves; a dihedral configuration; an elliptic Gorenstein leaf; a chain with exactly one non-invariant component of tangency order zero; and a star-shaped graph with non-invariant center of tangency order zero and $X$3-chain branches. Type (1) is terminal, and types (1)–(5) are canonical (Chen, 2021).

This classification places strong restrictions on possible exceptional configurations and is the basis for the surface theory of minimal log discrepancies. It also shows that foliated log canonicity is not simply a reformulation of the singularity theory of the ambient surface: the graph depends on foliation-specific data such as invariance, tangency order, and the $X$4-index.

A further refinement comes from adjoint singularities. For a foliated surface $X$5, one studies the adjoint divisor $X$6. For $X$7, the $X$8-adjoint log canonical singularities admit a complete classification. For $X$9, every $1$0-adjoint log canonical singularity is log canonical for $1$1, and for $1$2, every $1$3-adjoint canonical singularity is log canonical for $1$4; both bounds are sharp, as shown by explicit blow-up computations (Xu, 23 Dec 2025).

3. Threefold birational geometry and the foliated MMP

For corank-one foliations on $1$5-factorial normal projective threefolds, the log canonical MMP can be run under the standard assumption that $1$6 is klt. If $1$7 is foliated log canonical, then a $1$8-MMP exists and terminates with either a minimal model, where $1$9 is nef, or a Mori fiber space $K_{\mathcal F}$0 with $K_{\mathcal F}$1, $K_{\mathcal F}$2, and $K_{\mathcal F}$3 generated by curves tangent to $K_{\mathcal F}$4 on which $K_{\mathcal F}$5 is negative. In the Mori fiber space case, the general fibers are tangent to the foliation (Chaudhuri et al., 2024).

The foliated cone theorem has the same formal shape as in the classical MMP, but the negative extremal rays are generated by curves tangent to the foliation. Under the hypotheses above, there is a countable collection of rational curves $K_{\mathcal F}$6, all tangent to $K_{\mathcal F}$7, such that

$K_{\mathcal F}$8

with the underlying klt threefold length bound

$K_{\mathcal F}$9

Contractions of $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$0-negative exposed extremal rays exist, flips exist, and infinite sequences of $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$1-flips do not occur in dimension three (Chaudhuri et al., 2024).

Two structural features distinguish the foliated setting. First, the discrepancy inequalities are controlled by $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$2, so invariant and non-invariant exceptional divisors behave differently. Second, MMP steps are arranged around $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$3-tangent curves and preserve the foliation. These modifications are not cosmetic: they are what makes the lc category large enough to contain many natural examples, including foliations on $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$4 that are log canonical but not dlt (Chaudhuri et al., 2024).

Parallel rank-one results on threefolds focus on positivity rather than contraction theory. If $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$5 is a normal projective $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$6-factorial klt threefold and $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$7 is a rank-one log canonical foliated pair with $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$8, where $$\mathcal O_X(-K_{\mathcal F})\cong \det(\mathcal F)$$9 is ample and $\mathcal F$0, then nefness of $\mathcal F$1 implies semi-ampleness. In any dimension, if $\mathcal F$2 is $\mathcal F$3-factorial klt, $\mathcal F$4 is rank-one log canonical, and $\mathcal F$5, then in fact $\mathcal F$6 (Cascini et al., 3 Sep 2025).

4. Tangential log canonicity and the separatrix sector

A recent development isolates the tangential, or separatrix, sector of a foliated threefold pair. In a logarithmic simple adapted chart on a smooth threefold $\mathcal F$7, one fixes formal coordinates $\mathcal F$8, a reduced invariant normal crossing divisor

$\mathcal F$9

and a local generator

$\mathcal F\subset T_X$00

with positive non-resonance

$\mathcal F\subset T_X$01

The tangential locus is

$\mathcal F\subset T_X$02

Reduced tangential arcs are those $\mathcal F\subset T_X$03 with $\mathcal F\subset T_X$04 and $\mathcal F\subset T_X$05 (Corrêa, 2 Jul 2026).

The key confinement theorem states that, in the logarithmic simple adapted setting with positive non-resonance,

$\mathcal F\subset T_X$06

for every closed $\mathcal F\subset T_X$07. Equivalently, every reduced tangential arc centered on $\mathcal F\subset T_X$08 factors through $\mathcal F\subset T_X$09. This reduces the tangential arc geometry to the normalized separatrix–conductor system built from the normalizations $\mathcal F\subset T_X$10 of invariant branches and $\mathcal F\subset T_X$11 of pairwise conductors, glued by a seminormal pushout $\mathcal F\subset T_X$12 (Corrêa, 2 Jul 2026).

Foliated adjunction then transfers the discrepancy problem to ordinary log pairs on the normalized branches and conductors. For an invariant branch $\mathcal F\subset T_X$13, there is a canonically determined boundary $\mathcal F\subset T_X$14 on $\mathcal F\subset T_X$15 such that

$\mathcal F\subset T_X$16

The coefficients of $\mathcal F\subset T_X$17 are explicitly described: every other invariant trace has coefficient $\mathcal F\subset T_X$18, transverse boundary components retain the coefficients from $\mathcal F\subset T_X$19, and the normalization conductor different appears as well (Corrêa, 2 Jul 2026).

This leads to a tangential discrepancy

$\mathcal F\subset T_X$20

defined as the ordinary discrepancy of the normalized branch or conductor adjunction pair $\mathcal F\subset T_X$21. For toroidal invariant divisors read on branches, this tangential discrepancy agrees with the usual foliated discrepancy. The resulting arc-space theorem is a tangential version of the Ein–Mustaţă–Yasuda formula: $\mathcal F\subset T_X$22 From this one obtains a tangential inversion of adjunction, a cylinder criterion for tangential log canonicity, formulas for the tangential non-lc and non-klt loci, and lower semicontinuity of the toroidal tangential minimal log discrepancy (Corrêa, 2 Jul 2026).

A further refinement replaces ordinary discrepancies on the canonical image separatrix system by Mather–Jacobian discrepancies. If $\mathcal F\subset T_X$23 is the canonical image separatrix system on $\mathcal F\subset T_X$24, then

$\mathcal F\subset T_X$25

is model-independent, and one has the codimension formula

$\mathcal F\subset T_X$26

When the relevant stratum is l.c.i. and the Jacobian correction is trivial, the Mather–Jacobian and ordinary tangential discrepancies agree (Corrêa, 2 Jul 2026).

5. Minimal log discrepancies, thresholds, complements, and failures of naive analogies

The surface theory gives a precise description of foliated minimal log discrepancies. For a divisor $\mathcal F\subset T_X$27 over a surface germ, the foliated log discrepancy is $\mathcal F\subset T_X$28, and

$\mathcal F\subset T_X$29

If the minimal log discrepancy is negative, then it is $\mathcal F\subset T_X$30. For smooth SNC models with reduced foliation and coefficients $\mathcal F\subset T_X$31, Chen gives an explicit local formula for the mld in terms of which components are invariant, which intersections occur at smooth foliation points, and the coefficients of the boundary (Chen, 2021).

Two ACC theorems are known in low dimension. For foliated surface triples with coefficients in a DCC set $\mathcal F\subset T_X$32, the sets $\mathcal F\subset T_X$33 and $\mathcal F\subset T_X$34 satisfy ACC. More generally, for dimensions $\mathcal F\subset T_X$35 and ranks $\mathcal F\subset T_X$36, the sets of foliated log canonical thresholds

$\mathcal F\subset T_X$37

with coefficients in DCC sets $\mathcal F\subset T_X$38 and $\mathcal F\subset T_X$39 satisfy ACC. The proof uses foliated dlt modifications, adjunction to divisors over lc centers, and a finiteness theorem for coefficients of boundaries passing through lc centers (Chen, 2021, Chen, 2022).

On foliated surfaces one can go further. The set of mlds of lc rank-one foliated surface germs with coefficients in $\mathcal F\subset T_X$40 is

$\mathcal F\subset T_X$41

In particular, for empty boundary one gets

$\mathcal F\subset T_X$42

The same work proves boundedness of local complements, a local index theorem, uniform boundedness of mlds, and uniform rational lc polytopes for foliated surface germs (Liu et al., 2023).

These results also show where classical intuition fails. There are lc foliated surface germs with no $\mathcal F\subset T_X$43-complement, even though every lc rank-one foliated surface germ admits a $\mathcal F\subset T_X$44-complement. There are rational lc foliated surface germs that are not quotient singularities. Grauert–Riemenschneider type vanishing can fail for lc foliations on surfaces: an explicit example has

$\mathcal F\subset T_X$45

Such examples show that the foliated lc category is not merely a formal extension of lc pair theory (Liu et al., 2023).

6. Toric, toroidal, and log homogeneous realizations

In toric geometry, log canonical foliated pairs admit an especially explicit description. A toric foliation on a $\mathcal F\subset T_X$46-factorial toric variety $\mathcal F\subset T_X$47 corresponds to a complex vector subspace $\mathcal F\subset T_X$48, with rank $\mathcal F\subset T_X$49, and its canonical divisor is

$\mathcal F\subset T_X$50

A torus-invariant divisor $\mathcal F\subset T_X$51 is $\mathcal F\subset T_X$52-invariant if and only if $\mathcal F\subset T_X$53. For a toric foliated pair $\mathcal F\subset T_X$54 with $\mathcal F\subset T_X$55, the lc criterion becomes purely combinatorial: $\mathcal F\subset T_X$56 Equivalently, $\mathcal F\subset T_X$57 is lc if and only if the ordinary toric pair $\mathcal F\subset T_X$58 is lc (Fujino et al., 2024).

This toric reduction yields sharp birational consequences. If $\mathcal F\subset T_X$59, then every extremal ray $\mathcal F\subset T_X$60 of $\mathcal F\subset T_X$61 satisfies

$\mathcal F\subset T_X$62

If $\mathcal F\subset T_X$63, then the contraction of $\mathcal F\subset T_X$64 is a $\mathcal F\subset T_X$65-bundle and $\mathcal F\subset T_X$66. Fujita-type freeness and very ampleness follow in the expected toric range, and if $\mathcal F\subset T_X$67 is ample then

$\mathcal F\subset T_X$68

These are exact toric analogues of classical results, but for the foliated adjoint divisor $\mathcal F\subset T_X$69 (Fujino et al., 2024).

A broader bridge between foliations and ordinary lc pairs appears on toroidal and log homogeneous varieties. If $\mathcal F\subset T_X$70 is log canonical and there exists a Cartier divisor $\mathcal F\subset T_X$71 such that $\mathcal F\subset T_X$72 is locally free and globally generated, and if $\mathcal F\subset T_X$73 is a rank-one log canonical foliated pair, then there exists a reduced divisor $\mathcal F\subset T_X$74 such that $\mathcal F\subset T_X$75 is log canonical and

$\mathcal F\subset T_X$76

When $\mathcal F\subset T_X$77, this gives $\mathcal F\subset T_X$78 log canonical with

$\mathcal F\subset T_X$79

The divisor $\mathcal F\subset T_X$80 is constructed as a tangency divisor between $\mathcal F\subset T_X$81 and a general logarithmic distribution generated by global logarithmic vector fields (Spicer et al., 9 Apr 2026).

This construction effectively translates positivity, volume, and MMP questions for $\mathcal F\subset T_X$82 into the classical theory of lc pairs. On log homogeneous varieties it yields DCC for volumes of $\mathcal F\subset T_X$83, boundedness of canonical models, and an equivariant $\mathcal F\subset T_X$84-MMP. A plausible implication is that, in geometric settings with enough logarithmic vector fields, the birational behavior of a log canonical foliated pair can often be studied through an auxiliary ambient lc pair without losing the singularity control encoded by log canonicity (Spicer et al., 9 Apr 2026).