26 problems

Let $X$ be a Fano manifold of Picard number $1$, and let $f:X\to X$ be a surjective endomorphism. Here, $\deg f$ denotes the degree of $f$, and $\mathbb{P}^n$ denotes projective…

Coincidence conjecture. For every admissible sequence $\alpha$, the elements $e_\alpha$ and $\tilde e_\alpha$, respectively $f_{\alpha 0}$ and $\tilde f_{\alpha 0}$, coincide in…

Let $X$ be a smooth Fano variety of Picard number $\rho(X)=1$ over an algebraically closed field of characteristic zero. An endomorphism of degree greater than $1$ is a self-map of…

Conjecture on the non-prime density. The logarithmic density satisfies

Miyanishi's conjecture. Then $\phi$ is an isomorphism.

Sato's conjecture. $X$ is a projective space.

Let $K$ be a field of characteristic zero. The Dixmier conjecture. Every endomorphism of the first Weyl algebra … is an automorphism. Here $A_1(K)$ is generated by $t,x$ with…

Folklore conjecture. If $X$ admits a non-isomorphic surjective endomorphism, then $X$ is a projective space.

Let $X$ be a normal projective variety and let $f:X\to X$ be a non-isomorphic surjective endomorphism. The properties weakly $\delta$-imprimitive, strongly imprimitive, quasi-abeli…

Amerik–Rovinsky–Van de Ven conjecture. Then $X$ is isomorphic to projective space.

Let $X$ be a Fano manifold of Picard number $1$, meaning that $-K_X$ is ample and the Picard number of $X$ is $1$. Projective-space conjecture. If $X$ admits a non-isomorphic surje…

Linearity conjecture. Every totally invariant prime divisor of $\mathbb{P}^{n}$ is linear.

Involution-product conjecture. The endomorphism $u$ is the product of two involutions in $\operatorname{GL}(V)$ if and only if $u$ is similar to its inverse.

Exchange-property conjecture. The endomorphism $u$ is the sum of two square-zero endomorphisms of $V$ if and only if $u$ has the exchange property. Moreover, if…

Let $X$ be an algebraic variety over an algebraically closed field of characteristic zero, and let $\phi:X \longrightarrow X$ be an endomorphism. Let $Y$ be a proper closed subvari…

Let $X$ be a smooth Fano variety of Picard number one, meaning that $-K_X$ is ample and $\rho(X)=1$. Suppose that $X$ admits a non-isomorphic surjective endomorphism $f$. The proje…

Let $X$ be a normal projective variety over an algebraically closed field $k$ of characteristic zero. An endomorphism $f\colon X\to X$ is int-amplified if there exists an ample Car…

Test-element conjecture. Every test element of $H$ is a test element of $F_n$.

Surjective endomorphism conjecture. Every surjective endomorphism $f:X\to X$ must be bijective.

The first $g,h$ conjecture. For every endomorphism $f$ of $A_1$, there exist $g$ and $h$, both automorphisms, both anti-automorphisms or one is an automorphism and the other is an…

The first $\gamma,\delta$ conjecture. For every endomorphism $f$ of $A_1$, there exist involutions $\gamma$ and $\delta$, each of $\gamma$ and $\delta$ is conjugate to $\alpha$, su…

Let $X$ be a normal variety, and let $f\colon X\rightarrow X$ be an endomorphism of degree $\deg(f)>1$. Suppose that $X$ admits a log-canonical model … Let $E^{lc}_\mu$ be the sum…

Let $A$ be an absolutely simple abelian variety over a number field $K$. Let $\mathcal V$ be the set of finite places $v$ of $K$ for which $A$ has good reduction and…

Let $A_n$ be the $n$'th Weyl algebra over ${\mathbb C}$, let $A_n^{(t)}$ be the quantized Weyl algebra over the parameter ring used in the paper, and let $\operatorname{End}(A_n)$…

Nakayama's splitting conjecture. Under these hypotheses, $X\simeq Y\times E$ after a base change.