foliated log smooth JQZSC5TX

foliated log smooth

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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)