Foliated Minimal Model Program

• Foliated minimal model program (FMMP) is a framework extending classical MMP techniques to varieties with algebraic foliations, emphasizing flip termination and minimal model finiteness.
• It employs adaptations of divisorial contractions, flips, and the cone theorem to manage foliation-invariant curves and classify birational geometrics of foliated pairs.
• FMMP underpins applications ranging from toric examples to Mori dream spaces, paving the way for advances in the classification of complex algebraic foliations.

A foliated minimal model program (FMMP) is a framework for performing birational modifications of algebraic varieties equipped with algebraic foliations, extending the classical minimal model program (MMP) of higher-dimensional birational geometry to the setting where the canonical divisor is replaced by the canonical class of a foliation. For co-rank one foliations—those present as saturated, Lie-closed subsheaves of rank two in the tangent sheaf on a normal projective threefold—the FMMP now provides a definitive Mori-theoretic classification, including the existence and termination of flips, the structure of the cone of curves, and the finiteness and relations of minimal models. Results generalize to algebraically integrable, adjoint, and generalized foliated settings, and underpin the birational geometry of foliated pairs with log canonical singularities.

1. Foundations and Singularities in FMMP

A corank-one foliation $\mathcal{F}$ on a normal threefold $X$ is a saturated, Lie-closed subsheaf $\mathcal{F} \subset T_X$ of rank two, so that $T_X/\mathcal{F}$ is torsion-free of rank one. Its canonical class $K_\mathcal{F}$ is the divisor class corresponding to $\det \mathcal{F}^*$. The program considers foliated pairs $(\mathcal{F},\Delta)$, where $\Delta=\sum d_i D_i$ is a $\mathbb{Q}$-divisor with non-negative coefficients. The singularities of $(\mathcal{F},\Delta)$ are classified as foliated log canonical (F-lc) if, for any birational morphism $\pi:Y \to X$ with the pullback foliation $\mathcal{F}_Y=\pi^{-1}\mathcal{F}$, the discrepancies in

$$K_{\mathcal{F}_Y}+\Delta_Y = \pi^*(K_\mathcal{F}+\Delta)$$

are at least $-\epsilon(E)$, where $\epsilon(E)=0$ for $\mathcal{F}_Y$-invariant and $1$ otherwise. Similar thresholding defines F-dlt and F-terminal singularities. It is required that the ambient pair $(X,\Delta)$ is Kawamata log terminal (klt). In the boundary-polarized case, $\Delta=A+B$ has $A$ ample and $B \ge0$, providing an "ample part" for additional positivity control (Chaudhuri et al., 2024).

2. Core Theorems: Existence, Termination, and Outcomes

The FMMP for corank-one foliations on a $\mathbb{Q}$-factorial projective threefold establishes:

• For any F-lc pair $(\mathcal{F},\Delta)$ with $(X,\Delta)$ klt, the $(K_\mathcal{F}+\Delta)$-MMP—built from a sequence of divisorial contractions and flips—terminates.
• The output is either:
  1. A minimal model $(X_n,\mathcal{F}_n,\Delta_n)$ where $K_{\mathcal{F}_n}+\Delta_n$ is nef,
  2. Or a Mori fiber space $X_n\to Z$ with fibers tangent to $\mathcal{F}_n$ and $-(K_{\mathcal{F}_n}+\Delta_n)$ ample over $Z$.

• In the boundary-polarized case, if $K_\mathcal{F}+\Delta$ is pseudo-effective, the resulting minimal model is good: $K_{\mathcal{F}_n}+\Delta_n$ is semi-ample (Chaudhuri et al., 2024, Liu et al., 2024).

The structure of the FMMP parallels the classical MMP but includes strict control over the interaction of the foliation with the contracted and flipped loci to ensure invariance of the singularity type and integrability.

3. Foliated Cone and Contraction Theorems

The cone theorem describes the structure of the cone of curves:

$$\overline{NE}(X) = \overline{NE}(X)_{(K_\mathcal{F}+\Delta)\geq0} + \sum_{i} \mathbb{R}_+[C_i],$$

where $C_i$ are rational curves tangent to $\mathcal{F}$, satisfying $-6 \leq (K_\mathcal{F}+\Delta)\cdot C_i < 0$, and which generate the extremal rays (Chaudhuri et al., 2024, Spicer, 2017). This is a strict generalization of the Mori cone theorem, requiring that the negative rays involve only curves tangent to the foliation.

The contraction theorem ensures that any exposed $(K_\mathcal{F}+\Delta)$-negative extremal ray $R$ can be contracted via a morphism $c_R:X\to Y$ with relative Picard number $1$, contracting exactly those curves whose numerical classes lie in $R$. Depending on the locus contracted, these can be divisorial, fiber-type, or flipping contractions (Chaudhuri et al., 2024, Spicer, 2017).

4. Flips, Flops, and Termination

If a contraction is small (codimension at least $2$), then the flip exists and produces a new normal variety $X^+$ endowed with the induced foliation $(\mathcal{F}^+,\Delta^+)$ such that $K_{\mathcal{F}^+}+\Delta^+$ is ample over the base. The category of singularities and log canonicity is preserved under flips (Chaudhuri et al., 2024, Cascini et al., 2018). Termination of any sequence of flips is guaranteed by reduction to suitable dlt modifications and control over the intersection numbers $-6\leq (K_\mathcal{F}+\Delta)\cdot C$, ensuring the process cannot continue indefinitely.

Any two minimal models of a given $(\mathcal{F},\Delta)$ can be connected by a sequence of $(K_{\mathcal{F}_i}+\Delta_i)$-flops, generalizing the classical flop-connectivity results to the foliated setting. In boundary-polarized cases, the set of minimal models is finite, with the argument reduced to covering the space of divisors with finitely many rational polytopes, mimicking the Shokurov polytope method (Chaudhuri et al., 2024).

5. Extensions: Generalized, Integrable, and Adjoint FMMP

The FMMP extends to generalized foliated quadruples $(X,\mathcal{F},B,\mathbf{M})$ where $\mathbf{M}$ is a nef b-divisor encoding moduli or "nef-part" data, mirroring the framework for classical generalized pairs (Chen et al., 2023, Li, 30 Jun 2025). In the algebraically integrable case, the MMP is compatible with base change along the fibration whose fibers are the leaves of the foliation. For Q-factorial klt algebraically integrable adjoint foliated structures (with adjoint divisor $tK_\mathcal{F}+(1-t)K_X+B+\mathbf{M}_X$ for $t \in [0,1]$), the FMMP—run via cone, contraction, and flip theorems—terminates with either a good minimal model or a Mori fiber space (Cascini et al., 2024).

This generalized framework requires qdlt (quasi-divisorial log terminal) modifications and the use of canonical bundle formulas and adjunction to foliation-invariant divisors, in order to precisely control singularities and abundance phenomena.

6. Applications, Examples, and Broader Consequences

Applications and examples synthesize the theoretical structure:

• Toric foliated minimal model programs allow explicit combinatorial realization of every step of the program, encoding the data via Klyachko filtrations and determining discrepancies, contractions, and flips directly from the fan and filtration structure (Wang, 2022).
• For rank-one foliations on threefolds and surfaces, explicit FMMP runs illustrate the transition from complex rational to minimal models; characteristic examples include P$^3$ with linear foliations or Hirzebruch surfaces (Li, 30 Jun 2025, Wang, 2022).
• The program establishes that algebraically integrable lc Fano foliations on $\mathbb{Q}$-factorial klt varieties yield Mori dream spaces; canonical rings are finitely generated for polarized lc foliated triples (Liu et al., 2024).
• Via canonical bundle formulas compatible with the foliation, abundance and base-point-freeness results for semi-ample divisors (e.g., in the boundary-polarized case) hold, as shown for both the minimal model and Mori fiber space outcomes (Chaudhuri et al., 2024, Chen et al., 2023).
• Foliated Sarkisov program: for algebraically integrable or threefold foliations, any two Mori fiber spaces are connected by a sequence of Sarkisov links, paralleling the classical results for varieties (Chen et al., 21 May 2025).

7. Techniques, Open Questions, and Future Directions

Key techniques include F-dlt modifications (extracting exactly divisors with small discrepancies to reduce to F-dlt from F-lc), handling the failure of Bertini for foliations through recasting pairs as generalized quadruples, and the construction of Shokurov-type polytopes ensuring openness and finiteness phenomena in the space of boundaries (Chaudhuri et al., 2024, Chen et al., 2023, Liu et al., 2024). The canonical bundle formula for foliations is crucial in descending positivity along tangent Mori fiber spaces.

Open problems include extending the existence of contractions and flips beyond klt ambient spaces, the formulation of "pl-flips" for foliations, and improved singularity hierarchies distinguishing, for instance, “plt” and “terminal” for foliated structures (Liu et al., 2024). There is ongoing development for the FMMP in positive characteristic, Kähler settings, applications to moduli of foliated pairs, and the classification of algebraic foliations in higher codimension.

The current state of the FMMP establishes that, for a broad range of singular setting (F-lc, F-dlt, canonical, log-canonical), the birational geometry of corank-one foliations on threefolds matches that of ordinary varieties in all core respects: full MMP runs, termination, flop-connectedness, finiteness in boundary-polarized cases, and abundance in numerically trivial cases (Chaudhuri et al., 2024, Liu et al., 2024, Cascini et al., 2018, Chen et al., 2023, Cascini et al., 2024).