AN APPLICATION OF THE CANONICAL BUNDLE FORMULA

YURI G. PROKHOROV

Abstract. We prove a part of Shokurov’s conjecture on characterization of toric varieties modulo the minimal model program and adjunction conjecture.

1. Introduction

Let X be a proper toric variety and let D = ∑_{i=1}^{n} D_{i} be the invariant divisor. It is well-known that

n = dim X + rk(Weil divisors modulo numerical equivalence),

K_{X} + D ∼ 0, and the pair (X, D) has only log canonical singularities.

V.V. Shokurov proposed that these properties can characterize toric varieties: Conjecture 1.1 ([Sh]). Let (X/Z ∋ o, D = ∑ d_{i}D_{i}) be a log variety such that (X, D) has only log canonical singularities and −(K_{X} + D) is nef over Z. Then

(1.2)  ∑ d_{i} ≤ σ(X/Z) + dim X,

where σ(X/Z) is the rank of the group Weil divisors on X modulo the numerical equivalence over Z. Moreover, if the equality holds, then (X/Z ∋ o, ⌊D⌋) is a toric log pair, i.e, (X/Z ∋ o, ⌊D⌋) is formally (or analytically) isomorphic to a toric log pair. Example 1.3. If X/X ∋ 0 is a singularity germ, then σ(X/X) is exactly the rank of the group of Weil divisors modulo ℚ-Cartier divisors (see [K1]). For example, let X/X ∋ 0 be the singularity given by the equation xy+zt = 0 in ℂ^{4} and let D be the divisor cut out by xy = 0. Then (X, D) is log canonical, D has exactly four components and σ(X/X) = 1. Thus we have equality in (1.2). Clearly, the singularity X ∋ 0 is toric.

Conjecture 1.2 was proved by Shokurov in dimension 2 (see [Sh] and also [P1]). A special case of this conjecture in dimension 3 was verified in [P2]. In this paper we prove a weak form of Shokurov’s conjecture modulo the log minimal model program (LMMP) and the adjunction conjecture for fiber spaces. Before stating our main result we introduce notation and definitions.

This work was is carried out under the support of grants RFFI 02-01-00441 and INTAS-OPEN 2000-269.

Notation. We work over an algebraically closed field of characteristic zero. Notation and conventions of the minimal model theory [KMM], [Ut] will be used freely.

Let (X, D) be a proper log pair. Write D = ∑ d_{i}D_{i} where the D_{i} are irreducible components. We denote

∥D∥ = ∑ d_{i},

⟨D⟩ is the free abelian group generated by the D_{i},

⟨D⟩^{0} is the subgroup of ⟨D⟩ consisting of ℚ-Cartier numerically trivial divisors, σ(X, D) = rk (⟨D⟩/⟨D⟩^{0}) .

Let f: X → Z be a fiber type contraction. We say that a prime divisor P is horizontal if f(P) = Z and vertical if dim f(P) < dim Z. For any divisor D we have the decomposition D = D^{hor} + D^{vert} into the sum of horizontal and vertical parts.

Definition 1.4. Let (X, D) be a log pair. A numerical complement of K_{X} + D is a log divisor K_{X} + D′ with D′ ≥ D such that K_{X} + D′ has only log canonical singularities and numerically trivial. We say that a log divisor K_{X} +D is numerically complementary if it has at least one numerical complement.

Theorem 1.5. Let (X, D) be a proper log variety such that K_{X} + D is numerically complementary. Assume that in dimension dim X both the LMMP and Weak Adjunction Conjecture 2.1 hold. Then

(1.6)  ∥D∥ ≤ σ(X, D) + dim X.

Moreover, if the equality holds, then X is rational. We can omit the LMMP and Conjecture 2.1 as assumptions when dim X ≤ 3.

Corollary 1.7. Let X be a non-singular proper variety of dimension ≤ 3 and let D be a reduced simple normal crossing divisor on X such that K_{X}+D ≡ 0. Then ∥D∥ ≤ ρ(X) + dim X, where ρ(X) is the Picard number. Moreover, if the equality holds, then X is rational.

Remark. According to recent preprint [A2] Weak Adjunction Conjecture 2.1 hold in the case when K_{X} + D has a Kawamata log terminal numerical complement.

After finishing the main part of this work the author was informed that similar and even more general results were obtained by J. McKernan [M]. His proofs are completely different, much easier, and mainly do not use the LMMP.

Acknowledgments. The work has been completed during my stay at Max-Planck-Institut für Mathematik in August 2002. I would like to thank MPIM for hospitality and support. I also would like to thank Professor V.V. Shokurov for some remarks and Dr. O. Fujino, who informed me about [M].

2. Preliminary facts

Adjunction (see [K2], [A1], [F]). Let f: X → Z be a contraction and let D = ∑ d_{i}D_{i} be a ℚ-divisor on X such that d_{i} ≤ 1 whenever f(D_{i}) = Z.

For a prime divisor W ⊂ Z, define a number c_{W} as the log canonical threshold:

c_{W} = sup {c | (X, D + cf^{∗}W) is log canonical over the generic point of W } .

Then the ℚ-divisor

D_{Z} := ∑_{W}(1 − c_{W})W

is called the discriminant of K_{X} + D.

Note that the definition of the discriminant D_{Z} is a codimension one construction, so computing D_{Z} we can systematically remove codimension two subvarieties in Z and pass to generic hyperplane sections f_{H}: X ∩f^{−1}(H) → Z ∩ H. In particular, f^{∗}W is well-defined. Adjunction Conjecture 2.1 (weak form). Let f: X → Z be a fiber space and let D be a ℚ-divisor on X such that

(i) (X, D) is log canonical, (ii) K_{X} + D is ℚ-linearly trivial over Z.

Then there is an effective ℚ-divisor M_{Z} on Z such that K_{Z} + D_{Z} + M_{Z} is log canonical and

(2.2)  K_{X} + D ∼_{ℚ} f^{∗}(K_{Z} + D_{Z} + M_{Z}).

According to Kawamata [K2] this conjecture is true for contractions of relative dimension one.

Properties of σ. Lemma 2.3. Let g: X′ → X be a birational contraction, let D′ be the proper transform of D, and let E = ∑ E_{i} be the exceptional divisor. Then

(2.4)  σ(X′, D′ + E) ≤ σ(X, D) + ∥E∥.

Moreover, if g is a contraction of an extremal face in some LMMP, then in (2.4) the equality holds.

Proof. The surjective map g_{∗}: ⟨D′ + E⟩ → ⟨D⟩ induces the exact sequence

0 ⟶ ⟨E⟩ ⟶ ⟨D′ + E⟩/g^{∗}⟨D⟩^{0} ⟶ ⟨D⟩/⟨D⟩^{0} ⟶ 0

Since g^{∗}⟨D⟩^{0} ⊂ ⟨D′ + E⟩^{0}, this gives us the desired inequality. □

Corollary 2.5. Let g: X′ → X be a contraction of an extremal ray and let D be a divisor on X′. Then

σ(X′, D) − 1 ≤ σ(X, g_{∗}D) = σ(X′, D + E) − 1 ≤ σ(X′, D).

Furthermore, if E is a component of D, then σ(X, g_{∗}D) = σ(X′, D) − 1.

Easy results in the local case. Proposition 2.6 ([Ut, 18.22-23]). Let (X ∋ o, B) be a log canonical singularity such that every component of B is ℚ-Cartier. Then

∥B∥ ≤ dim X.

Moreover, if the equality holds, then X ∋ o is a quotient of a smooth point X′ by an abelian group 𝔄 acting on X′ free in codimension one. The proper transform B′_{i} of each component B_{i} ⊂ Supp(B) is 𝔄-stable. Therefore, (X, ⌊B⌋) is a ℚ-factorial toric log pair. Corollary 2.7. Let (X, B) be a log canonical log pair such that each component B_{i} of B is ℚ-Cartier. Then ∥B∥ ≤ codim ∩B_{i}. Corollary 2.8 (cf. [Ut, Corollary 18.24]). Let V be a projective ℚ-factorial log terminal variety with ρ(V) = 1 and let D be a boundary on V such that −(K_{V} + D) is nef and (V, D) is log canonical. Then ∥D∥ ≤ dim V + 1. Moreover, if the equality holds, then (V, ⌊D⌋) is a toric log variety. If V and all the D_{i} are defined over a non-closed field 𝕜, then V is 𝕜-rational.

Proof. Assume that

(2.9)  ∥D∥ ≥ dim V + 1

Take an embedding V ⊂ ℙ^{n} so that V is projectively normal and take a projective cone X ⊂ ℙ^{n+1} over V. Let B = ∑ b_{i}B_{i} be the corresponding cone over D. Then X is normal and ℚ-factorial. Let σ: X̃ → X be the blow up of the vertex, let B̃ be the proper transform of B, and let S be the exceptional divisor. We can write

σ^{∗}(K_{X} + B) = KX̃ + B̃ − a(S, B)S.

It is clear that S is a Cartier divisor and (S, B̃|_{S}) ≃ (V, D). Since K_{V} +D ≡ 0, we have a(S, B) = −1. Thus σ^{∗}(K_{X} + B) = KX̃ + B̃ + S and by the Inversion of Adjunction [Ut, 17.7] the pair (X̃, B̃+S) is log canonical. Hence so is (X, B). Moreover, (X̃, S) is plt (because V is log terminal). Now take (X′, B′) and 𝔄 such as in Proposition 2.6. Consider the diagram

\begin{matrix}X̃ & ⟵^{π̃} & X̃′ \\ ^{σ}↓ & & _{σ′}↓ \\ X & ⟵^{π} & X′ \end{matrix}

where X̃′ is the normalization of X̃ in the function field of X′. Let S′ = π̃^{−1}(S). Then π̃ is finite and π̃|_{X̃′\S′} is étale in codimension 1. By [Ut, 20.3] the pair (X̃′, S′) is plt and (X̃′, B̃′ +S′) is log canonical, where B̃′ = π̃^{−1}(B̃). In particular, S′ is irreducible and normal. Clearly each component of B̃′ is Cartier. Put Δ = B̃′|_{S′} and Δ_{i} = B̃′_{i}|_{S′}. Then K_{S′} + Δ is log canonical and numerically trivial, Δ = ∑ b_{i}Δ_{i}, ∑ b_{i} ≥ dim S′ + 1, and the Δ_{i} are ample numerically proportional Cartier divisors. Hence S′ is a Fano variety of (Fano) index ≥ dim S′ + 1 with only log terminal singularities. It is wellknown (see e.g. [IP, Th. 3.1.4]) that in this situation S′ ≃ ℙ^{N}, ∑ b_{i} = dim S′ + 1, and all the Δ_{i} are hyperplanes. This gives us the equality in (2.9). Further, V ≃ S = ℙ^{N}/𝔄, where 𝔄 acts on S′ = ℙ^{N} so that all the Δ_{i} are stable. By Corollary 2.7 we have ∩Δ_{i} = ∅ and ⌊Δ⌋ is a normal crossing divisor. This gives us that (V, ⌊D⌋) is a toric pair. For the last statement, we note that our construction is defined over 𝕜. Since ∩Δ_{i} = ∅, we can take 𝕜-coordinates on ℙ^{N} so that the action of 𝔄 is monomial. The statement is obvious in this case. □

Lemma 2.10. Let f: X → Z ∋ o be the contraction of an extremal ray R and let D = ∑ d_{i}D_{i} be a boundary on X such that all the components D_{i} are ℚ-Cartier and −(K_{X} + D) is f-nef. Assume that X is ℚ-factorial and log terminal, and (X, D) is log canonical. Put D^{hor} = ∑_{D_{i}·R>0} d_{i}D_{i} and D^{vert} = ∑_{D_{i}·R≤0} d_{i}D_{i}. Then ∥D^{vert}∥ ≤ codim f^{−1}(o). Moreover,

(i) if f is divisorial, then ∥D^{hor}∥ ≤ codim f(Exc(f)); (ii) if f is flipping and the flip χ: X −−→ X^{+} exists, then ∥D^{hor}∥ ≤

codim f^{+−1}(o) and ∥D∥ ≤ dim X + 1; (iii) if f is of fiber type and if the LMMP holds in dimensions < dim X,

then ∥D^{hor}∥ ≤ dim X/Z + 1. If furthermore the equality holds, then a generic fiber F is ℚ-factorial, ρ(F) = 1, the pair (F, ⌊D|_{F} ⌋) is toric, and X_{K} is K-rational, where K = K(Z) is the function field of Z.

Proof. (i) follows by Corollary 2.7.

(ii) Since χ(D^{hor}) ≤ χ(D)^{vert}, we have ∥D^{hor}∥ ≤ codim f^{+−1}(o). Thus ∥D∥ ≤ codim f^{−1}(o)+codim f^{+−1}(o). On the other hand, by [KMM, Lemma 5-1-17] we have codim Exc(f) + codim Exc(f^{+}) ≤ dim X + 1. This gives us ∥D∥ ≤ dim X + 1.

(iii) Let F be a generic fiber of f. Assume that ∥D^{hor}∥ ≥ dim X/Z + 1 = dim F + 1. Denote Δ = D|_{F} and Δ_{i} = D_{i}|_{F}. Then all the Δ_{i} are ample and numerically proportional. We claim that F is ℚ-factorial and ρ(F) = 1. Indeed, let g: F̃ → F be a small ℚ-factorialization and let Δ̃ = g^{∗}Δ. Then −(KF̃ + Δ̃) is nef, (F̃, Δ̃) is log canonical and F̃ has only log terminal singularities. Since Δ̃ ≠ 0, there is a KF̃-negative extremal ray R. Then Δ̃_{i}·R > 0 for all Δ̃_{i} = g^{∗}Δ_{i} (here we do not assume that Δ̃_{i} is irreducible). If ρ(F̃) > 1, this contradicts (i), (ii), or the inductive hypothesis. By Corollary 2.8 we have

∥D^{hor}∥ = ∥D|_{F} ∥ = dim F + 1

and the pair (F, ⌊D|_{F} ⌋) is toric. Since ∥D^{hor}∥ = ∥D|_{F} ∥, all the components of D are defined over K. So the last assertion follows by Corollary 2.8. □

3. Proof of Theorem 1.5

Let (X, D) be a log pair such that

(3.1)  ∥D∥ ≥ σ(X, D) + dim X.

Assume that K_{X} + D has a numerical complement K_{X} + G. Replace (X, G) with its minimal ℚ-factorial log terminal modification (blow-up divisors with discrepancy a(·, G) = −1) and D with the sum of its proper transform and the reduced exceptional divisor. Thus X is ℚ-factorial and log terminal. Run K_{X}-MMP. By Lemma 2.3 and Corollary 2.5 this preserves (3.1). All the divisorial contractions are positive with respect to G. Therefore, we cannot contract a connected component of G and K ≡ −G cannot be nef. At the end we get a fiber type extremal G-positive contraction f: X → Z. Note that our new X is ℚ-factorial and log terminal. It is sufficient to prove our theorem for this new X.

If Z is a point, then ρ(X) = 1 and X is a log terminal Fano variety . By Corollary 2.8, G = D and we have the equality in (3.1). Moreover, (X, ⌊D⌋) is a toric pair.

Consider the case when dim Z > 0. On this step we use the canonical bundle formula (2.2). Proposition 3.2. Assumptions as in Theorem 1.5. Assume additionally that X is ℚ-factorial, log terminal, and there exists a fiber type extremal contraction f: X → Z with dim Z > 0. Then the inequality (1.6) holds. Furthermore, if the equality holds, then K_{X} + D is numerically trivial over Z, a generic fiber F is ℚ-factorial, ρ(F) = 1, the pair (F, ⌊D|_{F} ⌋) is toric, and both X and Z are rational.

Proof. Assume (3.1). By Lemma 2.10 we have ∥D^{hor}∥ ≤ dim X/Z +1. Since f: X → Z is an extremal contraction, D^{vert} = f^{∗}Δ for some effective ℚ-Cartier divisor Δ. Write D_{Z} = D′_{Z} + D′′_{Z}, where D′_{Z} and D′′_{Z} are effective divisors without common components, Supp D′_{Z} ⊂ Supp Δ and none of the components of D′′_{Z} is contained in Supp Δ.

Claim 3.3. ∥D^{vert}∥ ≤ ∥D′_{Z}∥.

Proof. Let W ⊂ Z be a prime divisor and let S = f^{−1}(W)_{red}. Since ρ(X/Z) = 1, S is irreducible. Let d be the coefficient of S in D^{vert}. Write f^{∗}W = kS, where k ∈ ℕ. Then d + c_{W} k ≤ 1 (because (X, D + c_{W} f^{∗}W) is log canonical over the generic point of W). Hence, d ≤ 1 − c_{W}. This proves the statement. □

Assume that D^{hor} = 0. Then D^{vert} = D and

∥D^{vert}∥ ≥ σ(X, D) + dim X = σ(Z, Δ) + dim X.

On the other hand, by (2.2) we have that K_{Z} + G_{Z} + M_{Z} is log canonical and numerically trivial. It is clear that D′_{Z} ≤ D_{Z} ≤ G_{Z}, i.e., K_{Z} + G_{Z} + M_{Z} is a numerical complement of K_{Z} + D′_{Z}. Thus by the inductive hypothesis and Claim 3.3,

∥D^{vert}∥ ≤ ∥D′_{Z}∥ ≤ σ(Z, D′_{Z}) + dim Z,

a contradiction.

Now assume that D^{hor} ≠ 0. Then σ(X, D) ≥ σ(X, D^{vert})+1 and ∥D^{hor}∥ ≤ dim X/Z + 1 (see Lemma 2.10). Hence

(3.4) ∥D^{vert}∥ ≥ σ(X, D) + dim Z − 1 ≥

σ(X, D^{vert}) + dim Z ≥ σ(Z, Δ) + dim Z.

As above we have

(3.5)  ∥D^{vert}∥ ≤ ∥D′_{Z}∥ ≤ σ(Z, D′_{Z}) + dim Z.

This gives us the equalities in (3.4), (3.5) and (3.1). In particular, ∥D′_{Z}∥ = σ(Z, D′_{Z}) + dim Z. Since D′_{Z} is numerically complementary, the inductive hypothesis give us that Z is rational.

Finally, by Lemma 2.10 the pair (F, ⌊D|_{F} ⌋) is toric and X is rational. □

Since by [K2] Conjecture 2.1 holds when dim X/Z = 1, to prove the last part of the theorem we have to consider only the case when dim X = 3 and Z is a curve. In this case, Theorem 1.5 is an immediate consequence of the following. Proposition 3.6. Let f: X → Z be an extremal contraction to a curve, where X is ℚ-factorial and log terminal, and let D be a boundary on X such that K_{X} + D is numerically complementary. Then ∥D∥ ≤ dim X + σ(X, D). Furthermore, if the equality holds, then σ(X, D) = 2, K_{X} + D ≡ 0, and X is rational.

Proof. Let K_{X} + G be a numerical complement. Assume that

(3.7)  ∥D∥ ≥ σ(X, D) + dim X.

Since Z is a curve, ρ(X) = 2 and σ(X, D) ≤ 2. By Lemma 2.10,

(3.8)  ∥D^{hor}∥ ≤ ∥G^{hor}∥ ≤ dim X,  ∥D^{vert}∥ ≥ σ(X, D).

As ρ(X) = 2, the Mori cone \overline{NE}(X) is generated by two extremal rays, say R and Q. Let R is the ray corresponding to the curves in fibers of f.

Claim 3.9. There is a boundary Δ ≤ G such that (X, Δ) is a Kawamata log terminal log Fano variety.

Proof. Obviously, G^{vert}·Q > 0. Thus (K+G−εG^{vert})·Q < 0 for ε > 0. Since (K+G−εG^{vert})·R = 0, for 0 < ε_{1} ≪ ε, we have (K+G−εG^{vert}−ε_{1}G^{hor})·R < 0 and (K + G − εG^{vert} − ε_{1}G^{hor}) · Q < 0. Put Δ = G − εG^{vert} − ε_{1}G^{hor}. Then K + Δ is anti-ample and Kawamata log terminal. □

By the Cone Theorem the ray Q is contractible (i.e., there is a D^{vert}- positive extremal contraction g: X → W). This implies that Z ≃ ℙ^{1}. It is clear that g cannot have fibers of dimension ≥ 2. Since all the components of G^{vert} are strictly positive with respect to Q, by Lemma 2.10 we have

(3.10)  ∥D^{vert}∥ ≤ ∥G^{vert}∥ ≤ 2

If σ(X, D) = 1, then all the components of D are numerically proportional. By (3.8) and (3.7), D ≠ D^{hor}. Hence D = D^{vert}. This contradicts (3.10). Thus σ(X, D) = 2 and ∥D∥ ≥ dim X + 2. Combining (3.8), (3.10) and (3.7) we get ∥D^{vert}∥ = 2, ∥D^{hor}∥ = dim X, ∥D∥ = dim X + 2, D = G and K_{X} + D ≡ 0. The rest follows by Lemma 2.10. □

In conclusion, note that our method allow us to prove Conjecture 1.1 in the general form (at least in dimension three). We do not include these results because now there is much better approach [M].

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Department of Algebra, Faculty of Mathematics, Moscow State Lomonosov University, Moscow 117234, Russia

Max-Plank-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: prokhoro@mech.math.msu.su