8 problems

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

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

Let $X$ be a normal projective variety admitting a non-isomorphic polarized endomorphism. A normal projective variety is non-uniruled if it is not covered by rational curves, and i…

Let $X$ be a smooth projective variety of dimension $d$ over the algebraically closed field $\mathbb{k}$, 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 $f$…

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

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…