English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $X$ be a projective variety over an algebraically closed field $k$ ofarbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is$q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ andinteger $q > 1$.  Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. Weshow that the anti-canonical divisor $-K_X$ is numerically equivalent to aneffective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusionof Boucksom, de Fernex and Favre (Theorem C) and also covering singularvarieties over an algebraically closed field of arbitrary characteristic.  Suppose $f$ is separable and $X$ is normal. We show that the Albanesemorphism of $X$ is an algebraic fibre space and $f$ induces polarizedendomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$being pseudo-effective and $\mathbb{Q}$-Cartier means being a torsion$\mathbb{Q}$-divisor.  Let $f^{Gal}:\overline{X}\to X$ be the Galois closure of $f$. We show that if$p>5$ and co-prime to $deg\, f^{Gal}$ then one can run the minimal modelprogram (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for amildly singular threefold $X$ and reach a variety $Y$ with torsion canonicaldivisor (and also with $Y$ being a quasi-\'etale quotient of an abelian varietywhen $\dim(Y)\le 2$). Along the way, we show that a power of $f$ acts as ascalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when$X$ is a smooth and rationally chain connected projective variety of dimensionat most three.

Polarized endomorphisms of normal projective threefolds in arbitrary characteristic