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Let X be a normal compact Kähler space of dimension n. A surjective endomorphism f of such X is int-amplified if f * ξ − ξ = η for some Kähler classes ξ and η. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of X being smooth, a surface or a threefold with mild singularities, if X admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a Q-torus. Finally, we consider a normal compact Kähler threefold Y with only terminal singularities and show that, replacing f by a positive power, we can run the minimal model program (MMP) f -equivariantly for such Y and reach either a Q-torus or a Fano (projective) variety of Picard number one.

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We consider an arbitrary int-amplified surjective endomorphism f of a normal projective variety X over C and its f −1 -stable prime divisors. We extend the early result in [27, Theorem 1.3] for the case of polarized endomorphisms to the case of intamplified endomorphisms.Assume further that X has at worst Kawamata log terminal singularities. We prove that the total number of f −1 -stable prime divisors has an optimal upper bound dim X + ρ(X), where ρ(X) is the Picard number. Also, we give a sufficient condition for X to be rationally connected and simply connected. Finally, by running the minimal model program (MMP), we prove that, under some extra conditions, the end product of the MMP can only be an elliptic curve or a single point.2010 Mathematics Subject Classification. 14E30, 32H50, 08A35, Key words and phrases. amplified endomorphism, minimal model program, rationally connected variety.for any m > 0. The first equality is due to projection formula. By definition, the Iitaka dimension κ(X, F 1 ) = 0, a contradiction. Therefore, Claim 3.5 holds.By Claim 3.5, Equation (11) and (12), E 1 = E 2 = ∆ f = 0 and K X + j V j ∼ Q 0.Back to Equation (7), since ∆ f = 0, the ramification divisor consists of only these V j 's.By the purity of branch loci, f isétale outside ( V j ) ∪ f −1 (SingX), which completes the proof of Theorem 1.1 (4) for the case when dim Y 0 = 0. 142, arXiv:0408301.

Let X be a Q$\mathbb {Q}$‐factorial compact Kähler klt threefold admitting an action of a free abelian group G, which is of positive entropy and of maximal rank. After running the G‐equivariant log minimal model program, we show that such X is either rationally connected or bimeromorphic to a Q‐complex torus. In particular, we fix an issue in the proof of our previous paper [23, Theorem 1.3].