Toric Criterion
Intro
conjecture
complexity
complexity
Definition 1.1 Let X be a proper variety of dimension n, and let .X; / be a log pair. A decomposition of is an expression of the form X ai Si ; where Si 0 are Z-divisors and ai 0, 1 i k. The complexity of this decomposition is n C r d , where r is the rank of the vector space spanned by S1; S2; : : : ; Sk in the space of Weil divisors modulo algebraic equivalence and d is the sum of a1; a2; : : : ; ak. The complexity c D c.X; / of .X; / is the infimum of the complexity of any decomposition of .
absolute complexity
Definition 1.7 Let X be a proper variety of dimension n, and let .X; / be a log pair. The absolute complexity D .X; / of .X; / is n C d , where is the rank of the group of Weil divisors modulo algebraic equivalence and d is the sum of the coefficients of . If X is Q-factorial, then is the Picard number. (paper) (Brown et al., 2018, p. 925)
conjecture
Results
history
- in [2], Kollár proved the conjecture for Q-factorial germs of log canonical pairs; 2. in dimension two, McKernan and Keel proved the conjecture for projective surfaces of Picard rank one, [3]; Shokurov [1] proved the conjecture for arbitrary morphisms of surfaces; 3. in an unpublished note, Chelstov proved the conjecture for Q-factorial projective varieties of Picard number 1; 4. in [4], Prokhorov proved the conjecture for certain projective 3-folds. The method of his proof relies on the minimal model program in dimension three; 5. in [5], Yao gives a proof of the above conjecture for log smooth projective pairs (X, D) with KX +D ∼Q 0. Yao’s proof was inspired by the mirror-symmetry techniques of [6]; 6. finally, in [7], the authors settled the conjecture in the non-relative case, i.e., for a log canonical pair (X, B), where X is a proper variety - this setup corresponds to the relative setup where Z is just a point. (page) (Moraga and Svaldi, 2025, p. 3)
toric for surface germ
6.4. Theorem. Let (SIZ, B) have a log canonical K + Band neflZ divisor -(K + B). Then p(SIZ) ~ I:bi - 2, where p(SIZ) is the rank of the Weil group modulo the algebraic equivalencelZ or just the Picard number when the singularities of S are rational. Moreover, the equality holds if and only if K + B == 0 and SIZ is formally toric with C = LBJ ~ D. In addition, in the case with the equality and reduced B = C, (SIZ, C) is formally toric with C = D (see Example 5.3). (page) (Shokurov, 2000, p. 3922)
threefold toric
THEOREM 1.2. Let (X, D = ∑ diDi ) be a three-dimensional projective variety over C such that KX + D ≡ 0 and (X, D) has only purely log terminal singularities. Then (1.3) ∑ di ≤ rank(Weil(X)/≈) + 3 . Moreover, if the equality holds, then up to isomorphisms one of the following holds: (i) X P3, D = 0 or D = P2; (ii) X P1 × P2, D = 0 or D = {pt} × P2 or D = P1 × {line} (iii) X P(OP1 ⊕ OP1 ⊕ OP1 (d)), d ≥ 1, D is the section corresponding to the surjection OP1 ⊕ OP1 ⊕ OP1 (d) → OP1 (d); (iv) X P1×P1×P1, D = 0 or D = {pt}×P1×P1 or D = {pt1, pt2}×P1×P1; (v) X P(OP2 ⊕ OP2(d)), d ≥ 1, D is the negative section, or a disjoint union of two sections, one of them is negative; (vi) X P(OP1×P1 ⊕ L), L ∈ Pic(P1 × P1), D is the negative section, or a disjoint union of two sections, one of them is negative. In all cases (X, D ) is toric.
Precise theorem in BMSZ18 Main Thm A
Zotero
Main Thm A
THEOREM 1.2 Let X be a proper variety of dimension n, and let .X; / be a log canonical pair such that .KX C / is nef. If P ai Si is a decomposition of complexity c less than 1, then c 0 and there is a divisor D such that .X; D/ is a toric pair, where D and all but 2c components of D are elements of the set 1Si j 1 i ko. (paper) (Brown et al., 2018, p. 924)
Remarks
Cox ring of MDS
THEOREM 2.5.2 Let X be a normal Q-factorial projective variety such that Pic.X /Q D N 1.X /Q. Then X is a Mori dream space if and only if the Cox ring is finitely generated.
Sketch
step 1
Replace by dlt model.
dlt modification
PROPOSITION 2.2.3 Let .X; / be a log pair where X is a variety and the coefficients of belong to Œ0; 1 . Then there is a projective birational morphism W Y ! X such that (1) Y is Q-factorial, (2) only extracts divisors of log discrepancy at most 0, (3) if E D P Ei is the sum of the -exceptional divisors and is the strict transform of , then .Y; C E/ is divisorially log terminal and KY C E C D .KX C / C X a.Ei ;X;B/<0 a.Ei ; X; B/Ei : Any birational morphism W Y ! X satisfying (1)–(3) is called a divisorially log terminal modification. (paper) (Brown et al., 2018, p. 933)
step 2
Replace by MDS.
step 3
Show that the cox ring is a polynomial ring.
step 4
Deduct to original pair.
Cox ring
recall
local complexity
18.22 Theorem.
Let (X^^biBi) be log canonical at a point x G C\Bi. Assume that Kx and Bi are all Q-Cartier at x. Then ]P bi < dimX. In particular, Sn(local) has bounded sums.
pic 1 complexity
18.24 Corollary. Let X be an n-dimensional Fano variety with p(X) = 1 and let Y2 biBi be a Q-divisor such that Kx + ^2 b{B{ is lc and numerically trivial Then £ bi < dimX + 1. In particular, Sn(fano) has bounded sums.
Cox Ring of Toric
THEOREM 2.5.4 Let X be a normal Q-factorial projective variety of dimension n. Then X is toric if and only if the Cox ring is a polynomial ring generated by n C .X / variables, in which case the invariant divisors correspond to the coordinate hyperplanes. (page) (Brown et al., 2018, p. 947)
need to show
A local version
Lemma local version
LEMMA 2.4.3 Let .x 2 X; / be the germ of a log canonical pair where X has dimension n, and let P ai Di be a local decomposition. Assume that KX and D1; D2; : : : ; Dk are Cartier. If D n P ai D n d is the local complexity, then the following hold. (1) 0. (2) If < 1, then, after possibly reordering D1; D2; : : : ; Dk, there is an integer m n 2 0 such that .X; D1 C D2 C C Dm/ is log smooth and D1 C D2 C C Dm: (3) If < 3 2 , then either X is smooth at x or has a cAl singularity at x.
from local version to easy global version, using the projective cone and apply local version
r=1 lemma
Zotero
r=1 lemma
LEMMA 2.4.5 Let .X; / be a log canonical pair, where X is a projective variety of dimension n. Let P i k ai Si be a decomposition. If KX and S1; S2; : : : ; Sk all generate the same ray of the cone of effective divisors and this ray is also spanned by an ample divisor, then .
Step 3.1
r=1 implies
By some mmp and lemma above r=1 case
Zotero
r=1 case
LEMMA 3.3 Let X be a Q-factorial Kawamata log terminal projective variety of dimension n, and let .X; / be a log canonical pair. Let D D P ai Si be a decomposition of . If (1) KX C is numerically trivial, (2) d D Pk iD1 ai > n, and (3) S1; S2; : : : ; Sk all span the same ray of the cone of effective divisors, then the Picard number of X is 1.
induction on r
P1 bundle over MDS
Theorem 3.2. Let X be a Mori Dream Space. Choose D1, … , Dr Weil divisors generating the torsion free part of Cl(X), and let L be a nontrivial Cartier divisor which is in the subgroup generated by the Di . Let Y = PX(OX(L) ⊕ OX), which is a P1 bundle over X with projection π : Y → X. Then 1. The divisors π ∗Di, E∞, where E∞ is the section of X at infinity form a Z-basis for the torsion free part of Cl(Y ). 2. Cox(Y ; π ∗D1, … , π ∗Dr , E∞) ∼= Cox(X; D1, … , Dr )[s, t]. 3. Y is also a Mori Dream Space. (page) (Brown, 2013, p. 658)
smoothness of Cox ring
Cox lc if X CY type MDS
Zotero
Cox lc if X CY type MDS
Theorem 1.3. Let (X, ) be a projective Q-factorial log Calabi–Yau pair over C such that X is a Mori Dream Space, and let D1, … , Dr be a basis for the torsion free part of Cl(X). Then the ring Cox(X; D1, … , Dr ) is normal with log canonical singularities. (page) (Brown, 2013, p. 656)
CY and lc cox
Theorem 1.1. Let X be a Mori dream space over a eld of characteristic zero. Then X is of Calabi-Yau type if and only if the Cox ring of X has at worst log canonical singularities.
universal torsor
Proposition 2.2. Let X be a normal variety with Γ(X, O∗) = K∗ and finitely generated divisor class group. Let R be a sheaf of Cl(X)-graded algebras as constructed before, assume that the Cox ring R(X) is finitely generated, denote by p : ̂X → X the associated universal torsor and by X′ ⊆ X the set of smooth points. (i) The Cox ring R(X) is normal and every homogeneous invertible f ∈ R(X) is constant. (ii) The canonical morphism ̂X → X ̄ ̄ is an open embedding; in particular, ̂X is quasiaffine. (iii) The complement X ̄ ̄ \ p−1(X′) is small in X ̄ ̄, and the group H acts freely on the set p−1(X′) ⊆ ̂X. (iv) Every H-invariant Weil divisor on the total coordinate space X ̄ ̄ is principal. (page) (Hausen, 2008, p. 715)
MDS
fano type is MDS
LEMMA 2.5.3 Let X be a normal projective variety. The following are equivalent. (1) We may find a Kawamata log terminal pair .X; / such that .KX C / is ample. (2) We may find a Kawamata log terminal pair .X; / such that .KX C / is big and nef. (3) We may find a Kawamata log terminal pair .X; / such that KX C is numerically trivial and is big. In particular, if X is Q-factorial, then X is a Mori dream space.
Fano and CY type
Definition 2.6 (cf. [PS, Lemma-Definition 2.6]). Let X be a projective normal variety over a field, and let Δ be an effective Q-divisor on X such that KX + Δ is Q-Cartier. (i) We say that (X, Δ) is a log Fano pair if −(KX + Δ) is ample and (X, Δ) is klt. We say that X is of Fano type if there exists an effective Q-divisor Δ on X such that (X, Δ) is a log Fano pair. (ii) We say that X is of Calabi–Yau type if there exits an effective Q-divisor Δ such that KX + Δ ∼Q 0 and (X, Δ) is log canonical. (page) (Gongyo et al., 2014, p. 163)
small complexity implies fano type
Zotero
small complexity implies fano type
LEMMA 5.3 Assume Theorem 1.2n 1. Let X be a Q-factorial projective variety of dimension n. Suppose that .X; / is a divisorially log terminal pair and .KX C / is nef. If the complexity of .X; / is less than 1, then X is of Fano type. In particular, X is a toric variety. (page) (Brown et al., 2018, p. 957)
less numerical dimension
Zotero
less numerical dimension
LEMMA 4.4 Let .X; / be a divisorially log terminal pair, where X is a Q-factorial projective variety and .KX C / is nef. Let A0 be an ample divisor, and let 0 0 be a divisor such that KX C A0 C 0 is pseudoeffective, with numerical dimension k. Suppose that has a component P which is vertical for the ample model W X Z0 of KX C A0 C 0. Then there are an ample divisor A1 and a divisor 0 1 such that KX C A1 C 1 is pseudoeffective and the numerical dimension of KX C A1 C 1 is less than k. (page) (Brown et al., 2018, p. 952)
vertical component
LEMMA 4.3 Assume Theorem 1.2n 1. Let .X; / be a divisorially log terminal pair where X is a Q-factorial projective variety of dimension n. Let A be an ample divisor such that KX C A C is pseudoeffective, and let W X Z be the ample model of KX C A C . Assume that no component of N D N.X; KX C A C / is a component of . If the dimension of Z is greater than the complexity of .X; /, then we may find a component P of which is vertical. (page) (Brown et al., 2018, p. 951)
relative version
formally toric
Theorem 1. Let (X/Z, B) be a log canonical pair over a normal variety Z and let z ∈ Z be a closed point. Assume that −(KX + B) is nef over a neighborhood of z ∈ Z. Then, cz(X/Z, B) ≥ 0. Furthermore, if equality holds, then the following conditions are satisfied: 1. KX + B ∼Q,Z 0; 2. X → Z is formally toric at z; and, 3.. under the formal isomorphism of (2), the components of ⌊B⌋ are mapped to the completion of torus invariant divisors. (page) (Moraga and Svaldi, 2025, p. 2)
local formally toric
Theorem 2. Let x ∈ (X, B) be a log canonical singularity. Writing B = ∑︁ n i=1 biBi where the Bi are distinct prime divisors, then ∑︂n i=1 bi ≤ dim X + ρ(Xx). If equality holds, then (X, ⌊B⌋) is a formally toric pair at x.