Building blocks of amplified endomorphisms of normal projective varieties

Abstract

Let X be a normal projective variety. A surjective endomorphism f : X → X is int-amplified if f * L − L = H for some ample Cartier divisors L and H. This is a generalization of the so-called polarized endomorphism which requires that f * H ∼ qH for some ample Cartier divisor H and q > 1. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.2010 Mathematics Subject Classification. 14E30, 32H50, 08A35.

Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties

“…In this subsection, we first recall the definitions of polarized, amplified and int-amplified endomorphisms. Then, we refer to [18] for the general properties of int-amplified endomorphisms. Definition 2.3.…”

“…(2) f is amplified if f * D − D = H for some Cartier divisor D and ample Cartier divisor H; and [18]). Let f : X → X be an int-amplified endomorphism of a Qfactorial Kawamata log terminal projective variety X.…”

“…To make our symbols symmetric, we may denote by g 0 some power f t , so that g 0 descends to the int-amplified endomorphism g i of X i for each i. Lemma 2.6 (cf. [18]). Let f : X → X be an int-amplified surjective endomorphism of a projective variety X.…”

“…It is accessible under some good conditions for the variety X (cf. [18,Theorem 1.10]). See Theorem 2.4 for a detailed description.…”

“…The author would like to deeply thank Professor De-Qi Zhang for many inspiring ideas and discussions. Also, he would like to thank Doctor Sheng Meng for the result of running MMP [18] on int-amplified endomorphisms.…”