Recent progress on the Minimal Model Program for foliations

Abstract: We survey recent progress on the birational geometry of foliations on complex varieties. We focus on the MMP viewpoint: singularities, adjunction and applications to the MMP for foliations on surfaces and to the existence of flips on threefolds.

• The paper extends classical MMP techniques to singular foliations by adapting birational methods and addressing issues like non-resolvability of foliation singularities.
• It develops adjunction theory for foliations, linking canonical sheaves and log canonicity through innovative treatment of invariant and transverse cases.
• The study demonstrates the construction of flips in threefolds by reducing foliated flips to classical log flips, overcoming challenges from base point free limitations.

Recent Progress on the Minimal Model Program for Foliations

Introduction and Motivation

This essay surveys recent advances in the study of the Minimal Model Program (MMP) in the context of singular holomorphic foliations on complex varieties, as presented in "Recent progress on the Minimal Model Program for foliations" (2604.09027). The central thrust is the adaption and extension of birational techniques, well-established for varieties, to the setting where one studies pairs $(X, \mathcal F)$, with $X$ a complex variety (possibly singular) and $\mathcal F$ a coherent subsheaf of $T_X$ defining a foliation. Principal considerations involve the behavior of the canonical divisor $K_{\mathcal F}$ associated to the foliation, the structure of its singularities, adjunction theory, and the execution of the MMP both in surface and higher-dimensional cases, particularly investigating the construction of flips.

The analogy with the classical MMP for varieties is both fruitful and subtle: while one can, under mildness assumptions on foliation singularities, adapt much of the machinery (adjunction, cone theorems, contractions), the presence of genuinely new phenomena—such as non-resolvability of foliation singularities and the breakdown of base point free techniques—necessitates new tools and conceptual frameworks. The survey advances through the definition of singularities for foliated MMP, discusses adjunction for foliations, and systematically treats the MMP for foliations on surfaces and threefolds, highlighting structural results and remaining open problems.

Preliminaries and Foundations

Foliations and Canonical Sheaves

A foliation $\mathcal F$ of rank $r$ on a normal variety $X$ is defined as a saturated, bracket-closed coherent subsheaf $T_{\mathcal F} \subset T_X$. Association of a canonical sheaf $\omega_{\mathcal F} = (\wedge^r T_{\mathcal F})^*$ and a canonical divisor $X$0 is foundational; the behavior of $X$1 mirrors that of the canonical divisor for varieties but with crucial differences imposed by the transverse structure and singular behavior of leaves.

Singularities and Discrepancies

The singular locus of a foliation is characterized both algebraically (lack of local freeness of the normal sheaf, or, for derivations, via vanishing properties) and via Pfaffian maps. Discrepancies are defined analogously to the classical case, introducing an invariant $X$2 controlling birational geometry and facilitating singularities classification (canonical, log canonical, terminal) for foliated pairs $X$3.

Reduction versus Resolution

A salient obstruction unique to the foliated context is the impossibility of resolving all singularities to smooth foliations using blow-ups. Instead, one seeks reduction to well-understood distinguished singularities (e.g., canonical, simple), albeit the general theory is understood only in dimension two and three. The action of blow-ups on vector fields may generically worsen the singular behavior of the pullback foliation.

Singularities for Foliated MMP

Rank One and Co-rank One Foliations

For rank one foliations, log canonicity is characterized by the non-nilpotency of the linear part of the vector field in question, yielding an effective linear-algebraic criterion. Key implications include the openness of the log canonical locus for rank one foliations and the precise characterization of terminal points via invariance properties and holomorphic first integrals.

For co-rank one foliations, the notion of simple singularities plays a pivotal role. Simple singularities are tightly connected to formal normal forms and allow an explicit analysis of separatrices—a key ingredient in the structure theory required for the construction of flips.

Non-dicriticality and Algebraically Integrable Foliations

Non-dicriticality, which guarantees that exceptional divisors created in modifications remain invariant, is shown for canonical rank one foliations in all dimensions. For algebraically integrable foliations, the reduction of singularities problem is essentially toroidalization of the corresponding rational map, achieved via deep results extending to all dimensions.

Adjunction for Foliations

Adjunction theory for foliations provides linear equivalence formulas relating $X$4 restricted to a divisor $X$5 with $X$6 and a correction term $X$7. The theory distinguishes sharply between the invariant and transverse cases: in the non-invariant (transverse) case, log canonicity descends under adjunction, while in the invariant case, log canonicity may be lost on restriction, as demonstrated in explicit counterexamples.

Moreover, the difference $X$8 is interpreted via Pfaff fields and shown to carry strictly positive coefficients along divisors meeting the singular locus. This formalism is essential in subsequent applications of the MMP.

Minimal Model Program for Surface Foliations

Applying adjunction theory and the structure of singularities, the MMP for rank one foliations on surfaces is treated with parallelism to the classical case but significant subtleties. If the foliation is of canonical type and $X$9 is not pseudo-effective, the foliation is birational to a fibration by rational curves. If $\mathcal F$0 is pseudo-effective, a contraction sequence proceeds, guaranteed by careful control of discrepancies and the explicit adjunction formula.

A structural analysis for foliations of general type (big $\mathcal F$1) reveals that Zariski decompositions may support non-finite generation phenomena due to the existence of elliptic Gorenstein leaves—configurations of curves with vanishing intersection properties and complex dual graphs. This result demonstrates a critical divergence from the finite generation theorems central to the MMP for varieties.

Flips and Higher Dimensional Foliated MMP

The creation of flips in dimension three for foliations is a major technical advance. Due to the absence of a general base point free theorem, flip existence demands new strategies relying on the construction of suitable complements and the analysis of separatrices and meromorphic first integrals. Specifically, in the context of simple singularities or co-rank one canonical singularities on threefolds, the contraction of a $\mathcal F$2-negative extremal ray leads to exceptional loci covered by invariant rational curves, whose local neighborhoods can be understood via analytic or formal theory.

Finding a divisor $\mathcal F$3 such that the flipping locus is a log canonical center of $\mathcal F$4 allows the reduction of the foliated flip to a classical log flip; however, these divisors are generally dependent on the specific flipping contraction considered. The modularity and independence of such divisors in sequences of flips remains open.

Implications and Future Directions

The recent progress outlined in this survey underscores both the power and the limitations of extending birational techniques to singular foliations. Pragmatically, the adaptation of MMP machinery widens the landscape of classification problems amenable to reduction via flips and contractions in families of foliations. Theoretically, it exposes new pathologies absent in the case of varieties (notably non-finite generation of the canonical ring and the intricate structure of singularities under adjunction) and frames foundational questions—for example, the universality of non-dicriticality for canonical foliations of arbitrary rank, and the potential for more general base point freeness results in this context.

Anticipated future developments include deeper structural theorems for foliations of general type in higher dimensions, general criteria for the existence of log canonical complements independent of the flipping ray, and refined singularity theory for higher rank and higher codimension foliations. Progress in these domains will influence both the internal theory of foliations and their interaction with moduli spaces, algebraic dynamics, and arithmetic geometry.

Conclusion

This survey demonstrates that considerable progress has been made in developing an analogue of the Minimal Model Program for foliations, especially regarding singularities, adjunction, and the existence of flips in threefolds. Nevertheless, multiple fundamental challenges—most notably concerning the classification of singularities, the nature of adjunction, and the existence or independence of complements—remain central to ongoing research. The confluence of ideas from birational geometry, foliation theory, and formal geometry promises further substantial development in the near future.