12 problems

Broustet–Gongyo's conjecture. Every normal projective variety admitting a polarized endomorphism is of Calabi–Yau type. This has been proved for surfaces, smooth threefolds, and ra…

Gongyo's conjecture. After replacing $f$ by an iterate, the pair

Calabi–Yau type conjecture. Then $X$ is of Calabi–Yau type.

Let $X$ be a projective variety and let $f:X\to X$ be a polarized endomorphism, meaning that there is an ample divisor $H$ and a positive integer $q>1$ such that … A generically fi…

Let $X$ be a smooth projective variety of dimension $d$ over the algebraically closed field $\mathbbm{k}$, and let $f$ be a ($q$-)polarized endomorphism of $X$: there is an ample d…

Let $X$ be a normal projective variety, let $f\colon X\to X$ be a $q$-polarized endomorphism, and let $\Delta$ be an effective $\mathbb{Q}$-divisor such that $(X,\Delta)$ is an $f$…

A klt type variety is a variety that admits a klt pair structure. A polarized endomorphism is an endomorphism $f\colon X\rightarrow X$ for which $f^*A\sim mA$ for some ample diviso…

Folklore conjecture. The variety $X$ must be a finite quotient of a toric fibration over an abelian variety.

Broustet–Höring's conjecture. The morphism $\mu$ is an isomorphism in codimension one.

Broustet–Höring's conjecture. If $X$ is not necessarily $\mathbb{Q}$-Gorenstein, then $X$ has log canonical singularities.

Amerik's conjecture. If $X$ admits a polarized endomorphism, then

Let $X$ be a projective variety over $\mathbb{C}$ and let $\varphi:X\to X$ be an endomorphism with $\varphi^*L\cong L^{\otimes q}$ for some ample line bundle $L$ on $X$. Suppose th…