Toric foliation¶
Intro¶
toric foliation¶
toric foliated pair
Definition 3.1. A toric foliated pair (FW , ∆) consists of a toric foliation FW on a Q-factorial complete toric variety XΣ and a torus invariant R-divisor ∆ on XΣ whose coefficients belong to [0, 1]. (pdf) (Chang 和 Chen, 2023, p. 18)
Main results¶
foliation mmp:
toric foliated mmp
Theorem 0.4. Let F be a toric foliation with canonical singularities on a Q-factorial toric variety X. Then the MMP for F exists and ends with a foliation such that KF is nef or a fibration π : X → Z such that F is pulled back from a foliation on Z. Furthermore, F has non-dicritical singularities and the resulting foliation will also have non-dicritical singularities. (pdf) (Wang, 2023, p. 72)
foliated pair mmp:
fmmp for toric foliated pair md
fmmp for toric foliated pair
Theorem 0.8 (Propositions 4.1, 4.2, and 4.3)). Let (FW , ∆) be a toric foliated pair on a complete Q-factorial toric variety XΣ. Then FMMP works for (FW , ∆), and ends with (1) either with a toric foliated pair (G, ∆Y ) on YΣ′ such that KG + ∆Y is nef (2) or with a fibration π : X → Z and FW is pulled back from a foliation H on Z. Moreover, assuming the pair (FW , ∆) is log canonical, if FW is non-dicritical (resp. (FW , ∆) is F-dlt), then both G and H are non-dicritical (resp. both pairs (G, ∆Y ) and (H, π∗∆) are F-dlt). (pdf) (Chang 和 Chen, 2023, p. 2)
cone theorem:
cone for foliated toric
Corollary 4.9. Let (FW , ∆) be a log canonical toric foliated pair on a complete Q-factorial toric variety XΣ. Then NE(X)KFW +∆<0 = ∑ R≥0[Mi] where Mi’s are torus invariant rational curves tangent to FW . (pdf) (Chang 和 Chen, 2023, p. 26)
fibration:
fibration for pair
Theorem 1.1 (Lengths of extremal rational curves for toric foliated pairs). Let X be a projective (not necessarily Q-factorial) toric variety and let (F , ∆) be a log canonical toric foliated pair on X with rankF = r. Then l(F,∆)(R) := min [C]∈R{−(KF + ∆) · C} ≤ r + 1 holds for every extremal ray R of the Kleiman–Mori cone NE(X) = NE(X). Moreover, if l(F,∆)(R) > r holds for some extremal ray R of NE(X), then the contraction morphism φR : X → Y associated to R is a Pr-bundle over Y . In this case, F = TX/Y holds, where TX/Y is the relative tangent sheaf of φR : X → Y , and the sum of the coefficients of ∆ is less than one. In particular, the foliation F is locally free. (pdf) (Fujino 和 Sato, 2024, p. 1)
Why toric foliation?¶
Because we can¶
singularity¶
We need **foliated** log smooth resolution for foliated pairs.
defn fLS
Definition 2.6. We say that a pair of foliated (F , ∆ = ∑ i aiDi) on X is foliated log smooth if the following conditions hold: (1) (X, ∆) is log smooth. (2) F has only simple singularities. (3) Let S be the support of noninvariant components of ∆. For any closed point x ∈ S, we put Z as the minimal strata of Sing(F ) containing x. After re-indexing, we can assume that x ∈ Di and Di ⊂ S if and only if 1 ≤ i ≤ b. Then Z meets ⋂b i=1 Di transversally, that is dim Z ∩ ⋂b i=1 Di = dim Z − b. (pdf) (Chang 和 Chen, 2023, p. 13)
mld for fLS
Theorem 2.13. Let (F , ∆ = ∑ aiDi) be foliated log smooth of type one with ai ≤ 1. Suppose mld(F , ∆) ≥ 0. Then the following statements hold: (pdf) (Chang 和 Chen, 2023, p. 16)
defn fdlt
Definition 2.9. (F , ∆) is foliated divisorial log terminal (F-dlt) if (1) each irreducible component of ∆ is generically transverse to F and has coefficient at most one, and (2) there exists a foliated log resolution π : Y → X of (F , ∆) which only extracts divisors E of discrepancy > −ι(E). (pdf) (Chang 和 Chen, 2023, p. 14)
high dimensional algebraic integrable foliation¶
toroidal morphism
Definition 5.1.1 (cf. [ACSS21, Definition 2.1]). Let (X, ΣX , M)/U be a g-pair. We say that (X, ΣX , M) is toroidal if ΣX is a reduced divisor, M descends to X, and for any closed point x ∈ X, there exists a toric variety Xσ, a closed point t ∈ Xσ, and an isomorphism of complete local algebras φx : ̂OX,x ∼= ̂OXσ,t such that the ideal of ΣX maps to the invariant ideal of Xσ\Tσ, where Tσ ⊂ Xσ is the maximal torus of Xσ. Any such (Xσ, t) will be called as a local model of (X, ΣX , M) at x ∈ X. Let (X, ΣX , M)/U and (Z, ΣZ , MZ )/U be toroidal g-pairs and f : X → Z a surjective morphism/U . We say that f : (X, Σ, M) → (Z, Σ, MZ ) is toroidal, if for every closed point x ∈ X, there exist a local model (Xσ, t) of (X, ΣX , M) at x, a local model (Zτ , s) of (Z, ΣZ , MZ ) at z := f (x), and a toric morphism g : Xσ → Zτ , so that the diagram of algebras commutes. ̂OX,x ∼= / ̂OXσ,t ̂OZ,z ∼= / O ̂OZτ ,s O Here the vertical maps are the algebra homomorphisms induced by f and g respectively. (pdf) (Chen 等, 2023, p. 37)
equi-dimensional model
Definition-Theorem 5.1.2 ([LLM23, Definition-Theorem 6.5], [ACSS21, Theorem 2.2]). Let X be a normal quasi-projective variety, X → U a projective morphism, X → Z a contraction, B an R-divisor on X, M a nef/U b-divisor on X, D1, . . . , Dm prime divisors over X, and DZ,1, . . . , DZ,n prime divisors over Z. Then there exist a toroidal g-pair (X′, ΣX′ , M)/U , a log smooth pair (Z′, ΣZ′), and a commutative diagram X′ h / f′ X f Z′ hZ / Z satisfying the following. (1) h and hZ are projective birational morphisms. (2) f ′ : (X′, ΣX′ , M) → (Z′, ΣZ′) is a toroidal contraction. (3) Supp(h∗−1B) ∪ Supp Exc(h) is contained in Supp ΣX′ . (4) X′ has at most toric quotient singularities. (5) f ′ is equi-dimensional. (6) M descends to X′. (7) X′ is Q-factorial klt. (8) The center of each Di on X′ and the center of each DZ,i on Z′ are divisors. We call any such f ′ : (X′, ΣX′, M) → (Z′, ΣZ′) (associated with h and hZ ) which satisfies (1-7) an equi-dimensional model of f : (X, B, M) → Z. (pdf) (Chen 等, 2023, p. 37)
foliated log smooth
Definition 6.2.1 (cf. [ACSS21, §3.2]). Let (X, F, B, M)/U be a sub-gfq such that F is algebraically integrable. We say that (X, F, B, M) is foliated log smooth if there exists a contraction f : X → Z satisfying the following. (1) X has at most quotient toric singularities. (2) F is induced by f . (3) (X, ΣX ) is toroidal for some reduced divisor ΣX such that Supp B ⊂ ΣX . In particular, (X, Supp B) is toroidal, and X is Q-factorial klt. (4) There exists a log smooth pair (Z, ΣZ ) such that f : (X, ΣX , M) → (Z, ΣZ ) is an equi-dimensional toroidal contraction. (5) M descends to X. We say that f : (X, ΣX , M) → (Z, ΣZ ) is associated with (X, F, B, M), and also say that f is associated with (X, F, B, M). It is important to remark that f may not be a contraction/U . In particular, M may not be nef/Z. (pdf) (Chen 等, 2023, p. 45)
defn fLSR
Definition 6.2.3. Let X be a normal quasi-projective variety, B an R-divisor on X, M a nef/X b-divisor on X, and F an algebraically integrable foliation on X. A foliated log resolution of (X, F, B, M) is a birational morphism h : X′ → X such that (X′, F ′ := h−1F , B′ := h∗−1B + Exc(h), M) is foliated log smooth, where Exc(h) is the reduced h-exceptional divisor. (pdf) (Chen 等, 2023, p. 45)
has foliated log resulution md
has foliated log resulution
Lemma 6.2.4. Let X be a normal quasi-projective variety, B an R-divisor on X, M a nef/X b-divisor on X, and F a foliation on X that is induced by a dominant map f : X 99K Z. Then: (1) If f is a contraction, then for any equi-dimensional model f ′ : (X′, ΣX′ , M) → (Z′, ΣZ′) of f : (X, B, M) → Z associated with h : X′ → X and hZ : Z′ → Z, h is a foliated log resolution of (X, F, B, M) and h−1F is induced by f ′. (2) (X, F, B, M) has a foliated log resolution. (pdf) (Chen 等, 2023, p. 45)
fLS and lc
Lemma 6.2.2 (cf. [ACSS21, Lemma 3.1]). Let (X, F, B, M) be a sub-gfq such that F is algebraically integrable and (X, F, B, M) is foliated log smooth. Then (X, F, BF , M) is lc. (pdf) (Chen 等, 2023, p. 45)
toroidal to lc
Lemma 3.1. Let (X, ΣX ) and (Z, ΣZ) be toroidal pairs, let f : X → Z be a toroidal contraction and let F be the induced foliation. Let ∆ be the horizontal part of ΣX . Then (F , ∆) is log canonical. (pdf) (Ambro 等, 2022, p. 17)
Foliation on toric¶
Preliminaries¶
sheaves
equ qcoh sheaf and families md
equ qcoh sheaf and families
Proposition 1.10. [17, Theorem 4.9] The category of Σ-families is equivalent to the category of torus equivariant quasi-coherent sheaves on a smooth toric variety X. (pdf) (Wang, 2023, p. 75)
For any sheaf \(\mathcal{F}\) on \(X\), and cone \(\sigma\), there is a module \(F^{\sigma}\), and admits a \(T\)-action. Therefore it is a \(M\)-graded \(k[S_{\sigma}]\)-module \(F^{\sigma}=\oplus_{m\in M} F^{\sigma}_{m}\).
defn filtration of vec sp
Definition 2.5. A family of filtrations E is the data of a finite dimensional vector space E and for each facet F of P , an increasing filtration (EF (i))i∈Z of E such that EF (i) = {0} for i 0 and EF (i) = E for some i. We will denote by iF the smallest i ∈ Z such that EF (i) = 0. (pdf) (Clarke 和 Tipler, 2023, p. 392)
c1 of equ reflexive
Corollary 2.18. Let E = K(E) be an equivariant reflexive sheaf on X, given by a family of filtrations E = {(EF (i)) ⊂ E : F ≺ P, i ∈ Z}. The first Chern class of E is the class of the Weil divisor: c1(E) = − ∑ F ≺P iF (det E) DF . (2.5) where for all F ≺ P , iF (det E) = ∑ i∈Z ieF (i). (pdf) (Clarke 和 Tipler, 2023, p. 395)
local generator
local generators of toric foliation md
local generators of toric foliation
Lemma 1.7 ([Pan15, Lemma 2.1.12]). Let W be an r-dimensional complex vector subspace of NC and let Σ be a fan in NR. For any ray ρ ∈ Σ(1) with the primitive generator vρ, if ρ ⊂ W , then we choose v2, . . . , vn so that {vρ, v2, . . . , vr} forms a basis for W . Otherwise, we just choose {v1, . . . , vr} to be a basis for W . Then FW |Uρ is generated by δv1 , . . . , δvr if ρ 6⊂ W 1 χmρ δvρ, δv2 , . . . , δvr if ρ ⊂ W . (pdf) (Chang 和 Chen, 2023, p. 5)
Properties¶
invariant divisor
Lemma 3.4. Let F be a toric foliation on a Q-factorial toric variety X, which is determined by a subspace V of NC. Let Dρ be the toric divisor corresponding to the ray ρ. Then Dρ is F -invariant if and only if ρ ⊂ V . (pdf) (Wang, 2023, p. 82)
tangent curve
Lemma 3.3. Let ω be a codimension 1 cone in a simplicial fan Σ. Denote by Vω the vector space generated by the rays of ω. Then the torus invariant curve Dω is not tangent to the foliation FV if and only if V ⊂ Vω. (pdf) (Wang, 2023, p. 81)
tangent subvar
Proposition 4.7. Let FW be a toric foliation on a complete Q-factorial toric variety XΣ of dimension n with W 6= NC. Then for any cone τ ∈ Σ, Vτ is tangent to F if and only if W + Cτ = NC. (pdf) (Chang 和 Chen, 2023, p. 25)
F-invariant divisor
Proposition 1.12. (1) Let F1 and F2 be two foliations on an arbitrary normal variety X. If Z ⊂ X is a prime divisor that is invariant with respect to F1, then it is invariant with respect to F1 ∩ F2. (2) Let FW be a toric foliation on a toric variety XΣ. Then for any ρ ∈ Σ(1), Dρ is F -invariant if and only if ρ 6⊂ W . (pdf) (Chang 和 Chen, 2023, p. 7)
non-dicritical singularity
Definition 1.14. A foliation F of corank c on a normal variety X is called non-dicritical if for any divisor E over X with the dimension of cX (E) at most c − 1, each component of E is foliation invariant. F is dicritical if it is not non-dicritical. (pdf) (Chang 和 Chen, 2023, p. 8)
Singularity¶
description¶
dagger condition
Definition 1.18. Fix a lattice N. Let (Σ, W ) be a pair consisting of a fan Σ in NR and a complex vector subspace W ⊂ NC. A cone τ ∈ Σ is called non-dicritical if relint(τ ) ∩ W ∩ N 6= ∅ if and only if τ ⊂ W. We say (Σ, W ) satisfies the condition (†) if τ is non-dicritical for all τ ∈ Σ. (pdf) (Chang 和 Chen, 2023, p. 8)
fdlt and fls foliated toric md
fdlt and fls foliated toric
Proposition 0.5 (= Proposition 3.9). Let (FW , ∆) be a toric foliated pair on a Q-factorial toric variety XΣ. (1) (FW , ∆) is foliated log smooth if and only if Σ is smooth and (Σ, W ) satisfies the condition (†). Note that (FW , ∆) may not be log canonical. (2) (FW , ∆) is F-dlt if and only if the following statements hold true: (a) Supp(∆) ⊂ ⋃ ρ⊂W Dρ. (b) For any σ ∈ Σ satisfying φ(KFW +∆)|σ = 0, we have σ is smooth and non-dicritical. The latter means that either relint(σ) ∩ W ∩ N = ∅ or σ ⊂ W . (pdf) (Chang 和 Chen, 2023, p. 2)
log canonical etc foliated toric md
log canonical etc foliated toric
Proposition 0.4 (= Proposition 3.8). Let (FW , ∆) be a toric foliated pair on a Q-factorial toric variety XΣ. (1) (FW , ∆) is log canonical if and only if Supp(∆) ⊂ ⋃ ρ⊂W Dρ(= Supp(KFW )). (2) Suppose 0 < ε < 1. Then (FW , ∆) is ε-log canonical if and only if φ(KFW +∆)(u) ≥ ε for any primitive vector u ∈ |Σ| ∩ N such that R≥0u 6∈ Σ(1) where φ(KFW +∆) is the piecewise linear function associated with KFW + ∆. (3) FW is canonical if and only if for any σ ∈ Σ, the only non-zero elements of Πσ, W ∩ W ∩ N are contained in the facet of Πσ, W that does not contain the origin where Πσ, W is defined in Definition 3.6. (4) For any σ ∈ Σ, FW is terminal at the generic point of Vσ if and only if Πσ, W 6= σ and the elements of Πσ, W ∩ W ∩ N are vertices of Πσ, W . (pdf) (Chang 和 Chen, 2023, p. 2)
non-dicritical on smooth toric md
non-dicritical on smooth toric
Theorem 1.23. Suppose FW is a toric foliation on a smooth toric variety XΣ. Then the following statements are equivalent: (1) FW is non-dicritical. (2) FW is strongly non-dicritical. (3) (Σ, W ) satisfies the condition (†). (pdf) (Chang 和 Chen, 2023, p. 11)
relations¶
singularity of foliated toric pair md
singularity of foliated toric pair
Theorem 0.6. Let (FW , ∆) be a toric foliated pair on a Q-factorial toric variety XΣ. (1) (Proposition 2.11) If FW has only simple singularities, then it has at worst canonical singualrities. (2) (Corollary 3.10 and Proposition 3.11) If (FW , ∆) is F-dlt, then it is log canonical and FW is non-dicritical. (3) (Proposition 3.12) If (FW , ∆) is canonical, then FW is non-dicritical. (4) (Theorem 1.23 and Proposition 3.3) If XΣ is smooth, then the following statements are equivalent: (a) FW is non-dicritical. (b) FW is strongly non-dicritical (c) FW has only simple singularities. (pdf) (Chang 和 Chen, 2023, p. 2)
under MMP¶
pull back of foliation
Proposition 3.1. Let F be a toric foliation on a Q-factorial toric variety X, which is determined by a subspace V of NC. If ρ ⊂ V for some ρ ∈ Σ(1), then F is pulled back along some dominant rational map f : X Y with dim(Y ) < dim(X). In particular, if KF is not trivial, then F is a pull-back. (pdf) (Wang, 2023, p. 81)
Cone theorem¶
tangent extremal ray¶
decompose curve into eff and tangent class md
decompose curve into eff and tangent class
Proposition 3.5. Let F be a toric foliation on a Q-factorial toric variety X. Let C be a curve in X such that KF · C < 0. Then [C] = [M ] + α where M is a torus invariant curve tangent to the foliation and α is a pseudo-effective class. (pdf) (Wang, 2023, p. 82)
fibration¶
Q-factorial not pair:
fibration of foliation
Theorem 1.3 (Main theorem). Let X be a projective Q-factorial toric variety and let F be a toric foliation of rank r on X. Then lF (R) := min [C]∈R{−KF · C} ≤ r + 1 holds for every extremal ray R of NE(X) = NE(X). Moreover, if lF (R) > r holds for some extremal ray R of NE(X), then the contraction morphism φR : X → Y associated to R is a Pr-bundle over Y . In this case, F = TX/Y holds, where TX/Y is the relative tangent sheaf of φR : X → Y . In particular, F is locally free. (pdf) (Fujino 和 Sato, 2024, p. 622)
pair an non-Q-factorial:
lemma: toric projective bundles md
toric projective bundles
Lemma 6.1 (Toric projective bundles, [O1, p.41 Remark]). Let φ : X → Y be a toric morphism of toric varieties such that φ : X → Y is a Pr-bundle. Then X ≃ PY (L0 ⊕ · · · ⊕ Lr) for some line bundles L0, . . . , Lr on Y and φ : X → Y is isomorphic to the projection π : PY (L0 ⊕ · · · ⊕ Lr) → Y . (pdf) (Fujino 和 Sato, 2024, p. 10)