OSAMU FUJINO Throughout this note, we will work over C. We will freely use the basic definitions and results of the minimal model theory for log canonical pairs in [F1] and [F2].  The following theorems generalize Kawamata's famous result in [K]. Theorem 1 ([F5, Theorem 1.13]). Let (X, Δ) be a log canonical pair and let π: X → S be a projective morphism onto a variety S. Let R be a (K x + Δ)-negative extremal ray of NE(X/S) and let ϕ R : X → W  be the contraction morphism over S associated to R .  We put d := min dim E , where E runs over all positive-dimensional irreducible components of ϕ − 1 (P) for all P ∈ W . Then R is spanned by a possibly singular rational curve l with 0 < −(K x + Δ) · ℓ ≤ 2d More generally, we have: Theorem 2 ([F5, Theorem 1.12]). Let (X, Δ) be a log canonical pair and let φ: X → W be a projective morphism between varieties such that −( K x + ∆) is ϕ−ample. Let P be an arbitrary closed point of W . Let E be any positive-dimensional irreducible component of φ−1 (P). Then E is covered by (possibly singular) rational curves l with 0 < −(K x + ∆) · ℓ ≤ 2 dim E. In particular, E is uniruled. Our approach in [F5] is different from and is independent of Kawamata's in [K]. We think that the argument in [K] does not work for pairs whose singularities are worse than kawamata log terminal. Lemma 3 is a key lemma for the proof of Theorems 1 and 2. Lemma 3 ([F5, Lemma 12.1]). Let (X, Δ) be a log canonical pair and let ϕ : X → W  be a projective morphism. We take an arbitrary closed point P of W . Let E be any positive- dimensional irreducible component of ϕ− 1 (P). Let ν : E → E be tne normalization. Then, for every ample R-divisor H  on E, there exists an effective R-divisor Δ E,H  on E  such that ν*(K x + Δ) + H ～ R K E + Δ E , H + We note that Theorems 1, 2, and Lemma 3 are formulated for quasi-log schemes in [F5]. Hence the results in [F5] are much more general. Let us recall the definition of quasi-log schemes. For the details, see [F2, Chapter 6], [F4], and so on. Definition 4 (Quasi-log schemes). A quasi-log scheme is a scheme X endowed with an R- line bundle ω on X , a closed subscheme X −∞ ⊆ X , and a finite collection { C } of reduced and irreducible subschemes of X such that there is a proper morphism f : (Y, By) → X from a globally embedded simple normal crossing pair satisfying the following properties: (1) f*w ～ R Ky + By. (2) The natural map O x → f ∗ O y ([−(B ξ 1 )]) induces an isomorphism Ix_∞ = f ⋆ Oy([−(B$ 1 )] − [B$ 1 ]), where I x_∞ is the defining ideal sheaf of X_∞. (3) The collection of reduced and irreducible subschemes { C } coincides with the im- ages of the strata of (Y, By) that are not included in X_∞. We note that Y may be reducible in Definition 4 and that any log canonical pair has a natural quasi-log scheme structure. The notion of a basic slc-trivial fibration was first introduced in [F3] (see also [FH]). Definition 5 (Basic slc-trivial fibrations). A basic slc-trivial fibration f : ( X, B ) → Y consists of a projective surjective morphism f : X → Y and a simple normal crossing pair ( X, B ) satisfying the following properties: (1) Y is a normal variety, (2) every stratum of X is dominant onto Y and f ∗ O x ∼ O Y , (3) B is a Q-divisor such that B = B≤ 1 holds over the generic point of Y , (4) there exists a Q-Cartier Q-divisor D on Y such that K x + B ～ Q f*D , and (5) rankf*Ox([−(B<¹)]) = 1. We also note that X in Definition 5 is not necessarily irreducible.  The author thinks that the theory of quasi-log schemes is a powerful framework to use mixed Hodge structures on cohomology with compact support for the study of higher-dimensional algebraic varieties. He constructed the theory of basic slc-trivial fibrations in order to make the theory of variations of mixed Hodge structure on cohomology with compact support applicable for various geometric problems. Anyway, in [F5], we use quasi-log schemes and basic slc-trivial fibrations to establish Theorems 1, 2, and Lemma 3.
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[F1] O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789. [F2]  O. Fujino, Foundations of the minimal model program , MSJ Memoirs, 35. Mathematical Society of Japan, Tokyo, 2017. [F3]  O. Fujino, Fundamental properties of basic slc-trivial fibrations I, Publ. Res. Inst. Math. Sci. 58 (2022), no. 3, 473–526. O. Fujino, On quasi-log schemes, J. Math. Soc. Japan 75 (2023), no. 3, 829–856. [F4] O. Fujino, Cone theorem and Mori hyperbolicity, to appear in J. Differential Geom. F5 [FH] O. Fujino, K. Hashizume, Adjunction and inversion of adjunction, Nagoya Math. J. 249 (2023), 119–147. Y. Kawamata, On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609– [K] 611. DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, KYOTO UNIVERSITY, KYOTO 606-8502, Japan Email address: fujino@math.kyoto-u.ac.jp