Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds
Abstract
Let $f:X\to X$ be a non-isomorphic (i.e., $\text{deg } f>1$) surjective endomorphism of a smooth projective threefold $X$. We prove that any birational minimal model program becomes $f$-equivariant after iteration, provided that $f$ is $\delta$-primitive. Here $\delta$-primitive means that there is no $f$-equivariant (after iteration) dominant rational map $\pi:X\dashrightarrow Y$ to a positive lower-dimensional projective variety $Y$ such that the first dynamical degree remains unchanged. This way, we further determine the building blocks of $f$. As the first application, we prove the Kawaguchi-Silverman conjecture for every non-isomorphic surjective endomorphism of a smooth projective threefold. As the second application, we reduce the Zariski dense orbit conjecture for $f$ to a terminal threefold with only $f$-equivariant Fano contractions.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2023
- DOI:
- 10.48550/arXiv.2309.07005
- arXiv:
- arXiv:2309.07005
- Bibcode:
- 2023arXiv230907005M
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Dynamical Systems;
- Mathematics - Number Theory;
- 37P55;
- 14E30;
- 08A35
- E-Print:
- 48 pages, comments are welcome!