Compact Kähler threefolds with the action of an abelian group of maximal rank

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

“…(1) As mentioned in [Zho22, Remark 1.6], we were not able to run the G-MMP in [Zho22]. One of the difficulty to show the G-equivariancy we met was the unknown of the finiteness of (K X + ξ)-negative extremal rays where ξ is a nef and big class.…”

“…Remark 1.2 (Differences with the previous paper). Comparing the present paper with [Zho22], we give a few remarks on the differences of the approaches in the proofs.…”

“…In the analytic case, we showed in [Zho22] that, a Q-factorial compact Kähler terminal threefold X admitting an action of a free abelian group G, which is of positive entropy and of maximal rank, is either rationally connected or bimeromoprhic to a Q-complex torus, i.e., there is a finite surjective morphism from a complex torus T → X, which is étale in codimension one. In the proof of this main result [Zho22, Theorem 1.3] however, we applied the abundance theorem for Kähler threefolds; see [CHP16, recently claimed in [CHP21], the proof of [CHP16, Theorem 1.1] for the case when the Kodaira dimension κ(X) = 0 and the algebraic dimension a(X) = 0 seems incomplete.…”

“…In this note, we shall provide an independent proof of [Zho22,theorem 1.3] by constructing the G-equivariant minimal model program (G-MMP for short) for Kähler threefolds so as to fix the resulting issue in our previous paper. Moreover, thanks to the recent inspiring progress of the minimal model theory on Kähler threefolds ([DO22], [DH22];…”

“…We refer readers to [Zho22,Introduction] and the references therein for the background of this project. In this paper, we will focus on a clean proof of the existence of the G-MMP on Kähler threefolds (cf.…”