Log Canonical Foliations

• Log canonical foliations are holomorphic foliations on normal varieties paired with a boundary divisor, with singularities controlled like log canonical pairs in the minimal model program.
• They employ discrepancy computations and birational techniques—including divisorial contractions and flips—to establish minimal models and classify singularities through invariant and non-invariant divisors.
• Applications extend from surfaces to threefolds, offering new tools for moduli stability, characterization of foliation singularities, and advancements in birational geometry.

A log canonical foliation is a holomorphic foliation—i.e., a saturated, Lie-closed subsheaf of the tangent sheaf—on a normal complex (or, in positive characteristic, algebraic) variety, together with a boundary divisor, whose singularities are controlled in a way precisely analogous to the log canonical singularities of pairs in the minimal model program (MMP). The study of log canonical foliations, their singularities, birational geometry, and moduli, forms a fundamental component of the interface between foliation theory and higher-dimensional algebraic geometry, and plays a central role in extending the philosophy and techniques of the MMP to the foliated setting.

1. Formal Definitions and Singularities

Let $X$ be a normal complex projective variety. A holomorphic foliation $\mathcal{F}$ of rank $r$ is a saturated coherent subsheaf $\mathcal{F} \subset T_X$, closed under the Lie bracket, with generic rank $r$ (Cascini et al., 2023, Chen, 2021, Fan, 26 Dec 2025). For surface theory, one typically considers rank-1 foliations (subsheaves of the tangent bundle). To analyze singularities, one works with a foliated pair $(\mathcal{F}, \Delta)$, where $\Delta$ is a boundary divisor, and studies the behavior of the “foliated” canonical class $K_\mathcal{F}$, defined as the determinant of the dual $\mathcal{F}^\vee$.

The key invariant is the discrepancy: given a birational log resolution $\pi: Y \to X$ adapted to both the ambient singularities and the foliation, write

$$K_{\tilde{\mathcal{F}}} + \tilde{\Delta} = \pi^*(K_\mathcal{F} + \Delta) + \sum_E a_E(\mathcal{F}, \Delta) E,$$

where $E$ runs over $\pi$-exceptional divisors, $\tilde{\mathcal{F}}$ is the pull-back foliation, and $\tilde{\Delta}$ the strict transform plus reduced exceptional support. The pair $(\mathcal{F},\Delta)$ is log canonical (lc) if $a_E(\mathcal{F},\Delta) \geq -1$ for every such $E$ (Cascini et al., 2023, Chen, 2021, Chaudhuri et al., 2024).

In the surface case, the discrepancy is further refined to distinguish between invariant and non-invariant exceptional divisors, leading to nuanced threshold conditions for terminal, canonical, and log canonical singularities (Chen, 2021, Chen, 2021, Posva, 2023).

2. MMP and Minimal Models for Log Canonical Foliations

The foliated Minimal Model Program (MMP) adapts the classical apparatus—divisorial contractions, flips, minimal models, and Mori fiber spaces—to log canonical foliations. Constructions and principal results include:

• Existence of minimal models: If $X$ is a $\mathbb{Q}$-factorial klt projective variety, $\mathcal{F}$ is an algebraically integrable foliation of rank $r$, $\Delta \geq 0$, and $(\mathcal{F},\Delta)$ is lc, then under the hypothesis that flips terminate for klt pairs in dimension $r$, one obtains a finite sequence of foliated MMP steps ending with either a minimal model (where $K_{\mathcal{F}}+\Delta$ is nef) or a Mori fiber space (Cascini et al., 2023, Chaudhuri et al., 2024). The existence of such models underlies birational rigidity and adjunction phenomena.
• Foliated extremal rays and flip structure: Extremal rays in the cone of curves, divisorial contractions, and flips are defined with respect to the positivity of $K_{\mathcal{F}} + \Delta$, mirroring the classical setting. The technical machinery for constructing and terminating flips in the foliated context is directly derived from the MMP for varieties, with key additional input from the structure of foliation singularities and adaptions to the adjunction formula and discrepancy calculus (Chaudhuri et al., 2024).
• Adjunction and restriction: For an invariant divisor $D \subset X$, the restriction $$(K_{\mathcal{F}} + \Delta)|_D = K_{\mathcal{F}_D} + \Delta_D$$, where $\mathcal{F}_D$ is the restricted foliation, allows for inductive arguments and control of singularities under birational modifications (Cascini et al., 2023).

3. Classification of Singularities and Local Models

The singularities of log canonical foliations are classified via their exceptional divisors on minimal resolutions, with explicit dual graph types enumerated for surfaces (Chen, 2021, Chen, 2021):

• Terminal, canonical, and strictly lc types: Precise configurations such as chains of invariant rational curves, dihedral branches, cycles (elliptic Gorenstein leaves), and arrangements involving non-invariant curves define the local structure at singularities.
• Algorithmic resolution: By Seidenberg-type theorems, the resolution process for foliation singularities yields minimal models where the remaining singularities are of reduced “log canonical” type, distinguished by the discrepancies achieved on exceptional divisors.

For positive characteristic, the theory extends to 1-foliations closed under both the Lie bracket and the $p$-th power operation, with log canonicity formulated by requiring that discrepancies along non-invariant divisors are at least $-1$, and at least $0$ for invariant divisors (Posva, 2023).

4. Birational Positivity, Volume, and Stability Phenomena

The volumes of log canonical foliations, defined as $$\operatorname{vol}(\mathcal{F}, \Delta) = \operatorname{vol}(X, K_\mathcal{F} + \Delta)$$ in the sense of asymptotics of pluricanonical sections, satisfy strong finiteness and stability properties:

• Descending Chain Condition (DCC): For foliations birationally bounded by algebraically integrable families, the set of volumes $$\{ \operatorname{vol}(Y, K_\mathcal{G} + \Delta) \}$$ satisfies the DCC, precluding infinite strictly decreasing sequences (Fan, 26 Dec 2025). This is a foliated analogue of the well-orderedness statements for volumes of varieties of general type.
• Deformation invariance: For weak semistable families, the relative log canonical volume is independent of the parameter, extending the theory of invariance of plurigenera to the foliated setting (Fan, 26 Dec 2025).

Furthermore, for log canonical pairs with pseudo-effective canonical bundle, any quotient of the orbifold cotangent bundle has pseudo-effective determinant. Foliations with positive minimal slope with respect to a movable class are algebraically integrable with rationally connected leaves, per the Bogomolov–McQuillan criterion (Campana et al., 2015).

5. Applications: Surfaces, Threefolds, and Moduli

On surfaces, log canonical foliations possess bounded log canonical models; minimal partial du Val resolutions exist for all such models, and their moduli are bounded via numerical invariants such as the Hilbert function, the number of cuspidal components, and foliation indices (Chen, 2021). The functor parametrizing stable smoothable foliated surface pairs exhibits separateness, properness, and local-closedness in families, with plurigenera invariant for large multiples (Chen, 2021).

On threefolds, corank-one log canonical foliations admit fully developed MMP: existence and termination of flips, good minimal models in the boundary-polarized case, and finiteness plus flop-connectivity of minimal models. These results generalize classical theorems for varieties to the foliated case and provide new tools for the birational classification of Fano and Mori fiber space foliations (Chaudhuri et al., 2024).

6. Behavior Under Quotients and in Positive Characteristic

The interaction between log canonical foliations and group actions or inseparable quotients in positive characteristic is governed by preservation theorems: quotients by log canonical 1-foliations preserve terminal, canonical, klt, and lc singularities; in particular, for surface foliations with multiplicative singularities, the resulting quotient inherits klt properties (Posva, 2023).

For group actions induced by μ_p or general tamely ramified sources, the associated invariants (such as the foliation canonical divisor and discrepancies) transform compatibly with the ambient MMP structure, ensuring regularity of moduli and finiteness properties across a broad range of geometric settings.

7. Implications and Open Problems

The theory of log canonical foliations integrates foliation theory into the landscape of higher-dimensional algebraic geometry, leveraging the machinery of the MMP, vanishing theorems, and slope stability, with concrete impact in the classification and moduli of varieties and foliations. Open questions include extending the DCC property for volumes and multiplicities beyond algebraically integrable settings, the full invariance of plurigenera for families of foliated varieties, and the generalization of base-point-freeness and abundance conjectures within the foliated context (Fan, 26 Dec 2025).

The field continues to develop around key conjectures attaching finiteness, rigidity, and birational invariance properties to the log canonical regime, reflecting deep structural similarities and persistent contrasts with the established body of MMP for varieties.