Remarks on special kinds of the relative log minimal model program

Abstract

We prove R-boundary divisor versions of results proved by Birkar [B2] or Hacon-Xu [HX] on special kinds of the relative log minimal model program.

On minimal model theory for log abundant lc pairs

“…Similarly, we obtain κ σ (X/V, KX +∆−tḠ) = 0. Thus the morphisms (X,∆−tḠ) → V and V → Z satisfy all conditions of [25,Proposition 3.3]. By [25,Proposition 3.3], the pair (X,∆ − tḠ) has a good minimal model over Z for any t ∈ (0, t 0 ].…”

“…Since we haveψ * g * A = ψ * A and κ σ (Y, [15] for details). By [15, (3.3)], results of the minimal model theory proved with Lemma 2.11 (in particular, [21,Theorem 4.3] and results in [25]) can be recovered without any troubles.…”

“…By construction of lc closures, we have π c | X = π and π c−1 (Z) = X. Furthermore, we can construct (X c , ∆ c ) so that any lc center S c of (X c , ∆ c ) intersects X (see [25,Corollary 1.3]). Then the morphism (X c , ∆ c ) → Z c satisfies hypothesis of Theorem 1.1 because the relative invariant Iitaka dimension and the relative numerical dimension of R-Cartier divisors are determined by its restriction to a sufficiently general fiber of the Stein factorization.…”

“…By [25,Lemma 2.14], we can construct a sequence of steps of a (K X 1 + ∆ 1 )-MMP over Z with scaling of e 1 H 1…”

Non-vanishing theorem for $\mathrm{lc}$ pairs admitting a Calabi–Yau pair

“…Then a(P, W , Ψ ′ W ) = −1, and hence a(P, X, B ′ ) = −1 because (W , Ψ ′ W ) is a log smooth model of (X, B ′ ) (cf. [H2,Remark 2.11]). So P dominates Z by condition (3) in Step 1.…”