Int-amplified endomorphisms of compact Kähler spaces (Q2121815)

Let \(X\) be a smooth projective variety defined over an algebraically closed field of characteristic zero such that there exists a non isomorphic surjective endomorphism \(f : X \rightarrow X\). If \(\dim X=1\), then it follows from Hurwitz formula that \(X\) is either \(\mathbb{P}^1\) or an elliptic curve. In dimension 2, the surfaces which admit a non isomorphic surjective endomorphism have been classified by \textit{Y. Fujimoto} [Publ. Res. Inst. Math. Sci. 38, No. 1, 33--92 (2002; Zbl 1053.14049)] \textit{N. Nakayama} [Kyushu J. Math. 56, No. 2, 433--446 (2002; Zbl 1049.14029)]. Fano threefolds admitting a non isomorphic surjective endomorphism have been studied by \textit{S. Meng} et al. [Math. Ann. 383, No. 3--4, 1567--1596 (2022; Zbl 1502.14126)]. In this paper the author studied compact Kähler mainfolds admitting a non isomorphic surjective endomorphism. Let \(X\) be a normal compact Kähler space of dimension \(n\). A surjective endomorphism \(f\) of \(X\) is called int-amplified if \(f^{\ast}\xi -\xi = \eta\) for some Kähler classes \(\xi\) and \(\eta\). It is proved that when \(X\) is either smooth or 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. It is also shown that if \(X\) is a threefold with only terminal singularities, then replacing \(f\) by a positive power of it, it is possible to run a \(f\)-equivariant Minimal Model Program and reach either a Q-torus or a Fano variety of Picard number one.