On the log abundance for compact Kähler threefolds

Abstract

In this article we show that if (X, ∆) is a log canonical compact Kähler threefold pair such that K X + ∆ is nef and the numerical dimension ν(X, K X + ∆) = 2, then K X + ∆ is semi-ample. This result combined with our previous work in [DO23] shows that the log abundance holds for log canonical compact Kähler threefold pairs.

Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank

“…Since $$K_X+\Delta +\eta$$ is big by assumption, for a general fiber F of the MRC fibration, $$(K_X+\Delta +\eta )\cdot F>0$$. With the same argument as in the above paragraph after replacing [10, Theorem 10.12] by [9, Corollary 4.10], we finish the proof.$$\Box$$…”

“…First, we assume (1), that is, 𝐾 𝑋 + Δ 0 is pseudo-effective. Then, every (𝐾 𝑋 + Δ 0 + 𝛼)-negative extremal ray 𝑅 is (𝐾 𝑋 + Δ 0 )-negative, and hence generated by a rational curve 𝑅 = ℝ ≥0 [𝓁] (see [10,Theorem 10.12] and [5, Theorem 4.2]). Since (𝑋, Δ) is also klt with 𝐾 𝑋 + Δ being pseudo-effective, by [10, Theorem 10.12], there is a positive number 𝑑 such that every (𝐾 𝑋 + Δ + 𝜂)-negative extremal ray 𝑅 𝑖 with 𝑅 𝑖 = ℝ ≥ [𝓁 𝑖 ] satisfies −(𝐾 𝑋 + Δ) ⋅ 𝓁 𝑖 ≤ 𝑑 (note that here, we include the case (𝐾 𝑋 + Δ) ⋅ 𝓁 𝑖 ≥ 0).…”

“…Then, every $$(K_X+\Delta _0+\alpha )$$‐negative extremal ray R is $$(K_X+\Delta _0)$$‐negative, and hence generated by a rational curve $$R=\mathbb {R}_{\ge 0}[\ell ]$$ (see [10, Theorem 10.12] and [5, Theorem 4.2]). Since $$(X,\Delta )$$ is also klt with $$K_X+\Delta$$ being pseudo‐effective, by [10, Theorem 10.12], there is a positive number d such that every $$(K_X+\Delta +\eta )$$‐negative extremal ray $$R_i$$ with $$R_i=\mathbb {R}_{\ge }[\ell _i]$$ satisfies $$-(K_X+\Delta )\cdot \ell _i\le d$$ (note that here, we include the case $$(K_X+\Delta )\cdot \ell _i\ge 0$$). Therefore, …”

“…So, the abundance for numerically trivial pairs gives us $$K_Z+D_Z\sim _{\mathbb {Q}}0$$ (cf. [7, Corollary 1.18] or [10, Theorem 1.1]). We finish the proof of Claim 2.$$\Box$$…”

“…Let $$\phi : X\rightarrow Y$$ be the contraction of R (cf. [9, Theorems 1.5 and 2.18], [10, Section 10]). Then, G descends to a biregular action on Y and each $$\xi _i=\phi ^*\xi _{Y,i}$$ for some nef common $$G|_Y$$‐eigenvector $$\xi _{Y,i}$$ (cf.…”