--- title: Log Canonical Foliated Pairs url: https://www.emergentmind.com/topics/log-canonical-foliated-pairs type: topic --- # Log Canonical Foliated Pairs 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 [2104.00591; 2410.05178; 2607.01809]. ## 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 \(\mathcal F\) [2104.00591; 2410.05178]. For a proper birational morphism \(f:Y\to X\) with induced foliation \(f^*\mathcal F\), the discrepancy of a divisor \(E\) over \(X\) is defined by comparing \(K_{f^*\mathcal F}\) with the pullback of \(K_{\mathcal F}+\Delta\). In the McQuillan convention adopted in the surface and corank-one threefold literature, one sets \[ \varepsilon(E)= \begin{cases} 0 & \text{if }E\text{ is invariant for the pullback foliation},\\ 1 & \text{otherwise}, \end{cases} \] and defines \[ K_{\mathcal F_Y}+\Delta_Y=\pi^*(K_{\mathcal F}+\Delta)+\sum_E a(E,\mathcal F,\Delta)E. \] The pair is log canonical if \(a(E,\mathcal F,\Delta)\ge -\varepsilon(E)\) for every divisor \(E\) over \(X\), and klt if the inequality is strict. Canonical and terminal singularities are defined by the usual inequalities \(a(E,\mathcal F,\Delta)\ge 0\) and \(>0\) [2104.00591; 2410.05178; 2202.11346]. This invariant/transverse dichotomy is the first major difference from ordinary log pairs. Invariant divisors satisfy the stricter bound \(a(E,\mathcal F,\Delta)\ge 0\), whereas non-invariant divisors satisfy the classical-looking bound \(a(E,\mathcal F,\Delta)\ge -1\). 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 [2202.11346]. A variant convention appears in the rank-one threefold base-point-free theory: there the discrepancy formula is written with the strict transform of \(\Delta\) together with all non-\(\mathcal F_Y\)-invariant exceptional divisors added to \(\Delta_Y\) with coefficient \(1\), and the lc condition is then stated as \(a(E,\mathcal F,\Delta)\ge -1\). In that setup a basic consequence is that, if \((\mathcal F,\Delta)\) is log canonical, no irreducible component of \(\Delta\) is \(\mathcal F\)-invariant [2509.03109]. ## 2. Surface singularities and local structure On surfaces, log canonical foliated pairs admit a highly explicit local theory. If \(C\) is a non-invariant reduced curve on a smooth surface, the tangency order satisfies \[ K_{\mathcal F}\cdot C=\operatorname{tang}(\mathcal F,C)-C^2. \] If \(C\) is invariant, then one has the index identities \[ K_{\mathcal F}\cdot C=Z(\mathcal F,C)-\chi(C),\qquad C^2=\operatorname{CS}(\mathcal F,C), \] where \(Z\) is the local index along invariant curves and \(\operatorname{CS}\) is the Camacho–Sad index. At a non-degenerate singularity with local model \(\omega=\lambda y(1+o(1))dx-x(1+o(1))dy\), the two invariant branches satisfy \(Z=1\) and the Camacho–Sad residues are \(1/\lambda\) and \(\lambda\); at a saddle-node, the strong separatrix has \(Z=1\) and \(\operatorname{CS}=0\), while the weak separatrix, when present, carries the higher \(Z\)-index \(k+1\) [2104.00591]. 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 \((X,\mathcal F,p)\) 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 \(F\)-chain; a chain of three invariant curves with two \((-1)\)-\(F\)-curves and a bad tail; a chain of \((-2)\)-\(F\)-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 \(F\)-chain branches. Type (1) is terminal, and types (1)–(5) are canonical [2104.00591]. 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 \(Z\)-index. A further refinement comes from adjoint singularities. For a foliated surface \((X,\mathcal F)\), one studies the adjoint divisor \(K_{\mathcal F}+\epsilon K_X\). For \(\epsilon\in(0,1/3)\), the \(\epsilon\)-adjoint log canonical singularities admit a complete classification. For \(\epsilon\in(0,1/5)\), every \(\epsilon\)-adjoint log canonical singularity is log canonical for \(\mathcal F\), and for \(\epsilon\in(0,1/4)\), every \(\epsilon\)-adjoint canonical singularity is log canonical for \(\mathcal F\); both bounds are sharp, as shown by explicit blow-up computations [2512.20744]. ## 3. Threefold birational geometry and the foliated MMP For corank-one foliations on \(\mathbb Q\)-factorial normal projective threefolds, the log canonical MMP can be run under the standard assumption that \((X,\Delta)\) is klt. If \((\mathcal F,\Delta)\) is foliated log canonical, then a \(K_{\mathcal F}+\Delta\)-MMP exists and terminates with either a minimal model, where \(K_{\mathcal F'}+\Delta'\) is nef, or a Mori fiber space \(f:X'\to Y\) with \(\rho(X'/Y)=1\), \(\dim Y<\dim X\), and \(\operatorname{NE}(X'/Y)\) generated by curves tangent to \(\mathcal F'\) on which \(K_{\mathcal F'}+\Delta'\) is negative. In the Mori fiber space case, the general fibers are tangent to the foliation [2410.05178]. 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 \(\{C_i\}\), all tangent to \(\mathcal F\), such that \[ \overline{\mathrm{NE}}(X)=\overline{\mathrm{NE}}(X)_{(K_{\mathcal F}+\Delta)\ge 0}+\sum_i\mathbb R_{\ge 0}[C_i], \] with the underlying klt threefold length bound \[ -6\le (K_X+\Delta)\cdot C_i<0. \] Contractions of \(K_{\mathcal F}+\Delta\)-negative exposed extremal rays exist, flips exist, and infinite sequences of \(K_{\mathcal F}+\Delta\)-flips do not occur in dimension three [2410.05178]. Two structural features distinguish the foliated setting. First, the discrepancy inequalities are controlled by \(\varepsilon(E)\), so invariant and non-invariant exceptional divisors behave differently. Second, MMP steps are arranged around \(\mathcal F\)-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 \(\mathbb P^3\) that are log canonical but not dlt [2410.05178]. Parallel rank-one results on threefolds focus on positivity rather than contraction theory. If \(X\) is a normal projective \(\mathbb Q\)-factorial klt threefold and \((\mathcal F,\Delta)\) is a rank-one log canonical foliated pair with \(\Delta=A+B\), where \(A\) is ample and \(B\ge 0\), then nefness of \(K_{\mathcal F}+\Delta\) implies semi-ampleness. In any dimension, if \(X\) is \(\mathbb Q\)-factorial klt, \((\mathcal F,\Delta)\) is rank-one log canonical, and \(K_{\mathcal F}+\Delta\equiv 0\), then in fact \(K_{\mathcal F}+\Delta\sim_{\mathbb Q}0\) [2509.03109]. ## 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 \(W\), one fixes formal coordinates \(x_1,x_2,x_3\), a reduced invariant normal crossing divisor \[ \mathscr D=\{x_1\cdots x_r=0\},\qquad 1\le r\le 3, \] and a local generator \[ \omega=u\,x_1\cdots x_r\left(\sum_{i=1}^r\lambda_i\frac{dx_i}{x_i}+\sum_{j=r+1}^3 h_j(x)\,dx_j\right), \] with positive non-resonance \[ a_1\lambda_1+\cdots+a_r\lambda_r\ne 0 \quad\text{for every }(a_1,\ldots,a_r)\in\mathbb Z_{\ge 0}^r\setminus\{0\}. \] The tangential locus is \[ \Sigma_{\tan}:=\operatorname{Sing}(\mathcal G)\cap |\mathscr D|. \] Reduced tangential arcs are those \(\gamma\in J_\infty(W)_{\mathrm{red}}\) with \(\gamma(0)\in Z\) and \(\gamma^*\omega=0\) [2607.01809]. The key confinement theorem states that, in the logarithmic simple adapted setting with positive non-resonance, \[ J_\infty^{\tan}(W,\mathcal G;Z)_{\mathrm{red}}=J_\infty(\mathscr D;Z)_{\mathrm{red}} \] for every closed \(Z\subset \mathscr D\). Equivalently, every reduced tangential arc centered on \(\mathscr D\) factors through \(\mathscr D\). This reduces the tangential arc geometry to the normalized separatrix–conductor system built from the normalizations \(S_\alpha^\nu\) of invariant branches and \(C_{\alpha\beta}^\nu\) of pairwise conductors, glued by a seminormal pushout \(\mathscr S_W^{sn}\) [2607.01809]. Foliated adjunction then transfers the discrepancy problem to ordinary log pairs on the normalized branches and conductors. For an invariant branch \(S\), there is a canonically determined boundary \(\Theta_S\) on \(S^\nu\) such that \[ \nu^*\big((K_{\mathcal G}+\Delta_W)|_S\big)\sim_{\mathbb Q}K_{S^\nu}+\Theta_S. \] The coefficients of \(\Theta_S\) are explicitly described: every other invariant trace has coefficient \(1\), transverse boundary components retain the coefficients from \(\Delta_W\), and the normalization conductor different appears as well [2607.01809]. This leads to a tangential discrepancy \[ a_{\tan}(F;X,\mathcal G,\Delta):=a(F;V,B_V), \] defined as the ordinary discrepancy of the normalized branch or conductor adjunction pair \((V,B_V)\). 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: \[ lcodim_{\tan}\bigl(N_q^{\tan}(E)\bigr)=q\,a_{\tan}(E;X,\mathcal G,\Delta). \] 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 [2607.01809]. A further refinement replaces ordinary discrepancies on the canonical image separatrix system by Mather–Jacobian discrepancies. If \(V_X\) is the canonical image separatrix system on \(X\), then \[ a^{\tan}_{MJ}(E;X,\mathcal G,\Delta)=a_{MJ}(F;V_X,{}_{V_X}) \] is model-independent, and one has the codimension formula \[ lcodim^{MJ}_{\tan}\bigl(N_q^{\tan}(E)\bigr)=q\,a^{\tan}_{MJ}(E;X,\mathcal G,\Delta). \] When the relevant stratum is l.c.i. and the Jacobian correction is trivial, the Mather–Jacobian and ordinary tangential discrepancies agree [2607.01809]. ## 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 \(E\) over a surface germ, the foliated log discrepancy is \(a(E,\mathcal F,\Delta)+\varepsilon(E)\), and \[ \operatorname{mld}(x;\mathcal F,\Delta) =\inf\{a(E,\mathcal F,\Delta)+\varepsilon(E)\mid c_X(E)=x\}. \] If the minimal log discrepancy is negative, then it is \(-\infty\). For smooth SNC models with reduced foliation and coefficients \(\le 1\), 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 [2104.00591]. Two ACC theorems are known in low dimension. For foliated surface triples with coefficients in a DCC set \(B\subset[0,1]\), the sets \(PLD(2,B)\) and \(MLD(2,B)\) satisfy ACC. More generally, for dimensions \(n\le 3\) and ranks \(r0}\) 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 [2104.00591; 2202.11346]. On foliated surfaces one can go further. The set of mlds of lc rank-one foliated surface germs with coefficients in \(\Lambda\subset[0,1]\) is \[ \left\{0,\ \frac{1-\sum c_i\gamma_i}{n}\ \middle|\ n\in\mathbb N^+,\ c_i\in\mathbb N,\ \gamma_i\in\Lambda\right\}\cap[0,1]. \] In particular, for empty boundary one gets \[ \left\{\,0,\ \frac1n\ \middle|\ n\in\mathbb N^+\,\right\}. \] 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 [2305.06493]. These results also show where classical intuition fails. There are lc foliated surface germs with no \(1\)-complement, even though every lc rank-one foliated surface germ admits a \(2\)-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 \[ R^1f_*\mathcal O_S(K_{\mathcal F})\ne 0. \] Such examples show that the foliated lc category is not merely a formal extension of lc pair theory [2305.06493]. ## 6. Toric, toroidal, and log homogeneous realizations In toric geometry, log canonical foliated pairs admit an especially explicit description. A toric foliation on a \(\mathbb Q\)-factorial toric variety \(X(\Sigma)\) corresponds to a complex vector subspace \(V\subset N_\mathbb C\), with rank \(\dim_\mathbb C(V)\), and its canonical divisor is \[ K_{\mathcal F_V}=K_X+\sum_{\rho\in V}D_\rho. \] A torus-invariant divisor \(D_\rho\) is \(\mathcal F_V\)-invariant if and only if \(\rho\notin V\). For a toric foliated pair \((\mathcal F,\Delta)\) with \(\Delta=\sum b_\rho D_\rho\), the lc criterion becomes purely combinatorial: \[ (\mathcal F,\Delta)\text{ is lc} \iff \operatorname{Supp}\Delta\subset \operatorname{Supp}K_{\mathcal F} \text{ and all }b_\rho\in[0,1]. \] Equivalently, \((\mathcal F,\Delta)\) is lc if and only if the ordinary toric pair \((X,\sum_{\rho\notin V}D_\rho+\Delta)\) is lc [2410.17009]. This toric reduction yields sharp birational consequences. If \(r=\operatorname{rank}\mathcal F\), then every extremal ray \(R\) of \(\overline{NE}(X)\) satisfies \[ l_{(\mathcal F,\Delta)}(R)\le r+1. \] If \(l_{(\mathcal F,\Delta)}(R)>r\), then the contraction of \(R\) is a \(\mathbb P^r\)-bundle and \(\mathcal F=T_{X/Y}\). Fujita-type freeness and very ampleness follow in the expected toric range, and if \(L-(K_{\mathcal F}+\Delta)\) is ample then \[ H^i(X,\mathcal O_X(L))=0\qquad\text{for every }i>0. \] These are exact toric analogues of classical results, but for the foliated adjoint divisor \(K_{\mathcal F}+\Delta\) [2410.17009]. A broader bridge between foliations and ordinary lc pairs appears on toroidal and log homogeneous varieties. If \((X,B)\) is log canonical and there exists a Cartier divisor \(D\) such that \(T_X(-\log B)\otimes\mathcal O(D)\) is locally free and globally generated, and if \((\mathcal F,\Delta)\) is a rank-one log canonical foliated pair, then there exists a reduced divisor \(\Gamma\) such that \((X,\Delta+\Gamma)\) is log canonical and \[ K_X+\Delta+\Gamma\sim K_{\mathcal F}+\Delta+D. \] When \(\Delta=0\), this gives \((X,\Gamma)\) log canonical with \[ K_X+\Gamma\sim K_{\mathcal F}+D. \] The divisor \(\Gamma\) is constructed as a tangency divisor between \(\mathcal F\) and a general logarithmic distribution generated by global logarithmic vector fields [2604.08100]. This construction effectively translates positivity, volume, and MMP questions for \(K_{\mathcal F}\) into the classical theory of lc pairs. On log homogeneous varieties it yields DCC for volumes of \(K_{\mathcal F}+\Delta\), boundedness of canonical models, and an equivariant \(K_{\mathcal F}+\Delta\)-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 [2604.08100]. Source: https://www.emergentmind.com/topics/log-canonical-foliated-pairs