43 problems

Work over an algebraically closed field $k$ of characteristic zero, and let varieties over $k$ mean separated integral schemes of finite type over $k$. Let $\varphi:X\to X$ be an e…

Irreducibility conjecture. For each $n\ge 3$, there exists a value of $c$ such that $C_f$ is irreducible.

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

Generalized Dixmier conjecture. Every endomorphism of $A_n(K)$ is an automorphism. This extends the first-Weyl-algebra problem to all Weyl algebras; the supplied text gives no reso…

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

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

Point-evaluation conjecture. Every nonzero compact endomorphism of $A$ is essentially a point evaluation.

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

Let $G_{k,n}$ be the complex Grassmann manifold, and let $H^{*}(G_{k,n};\mathbb{Q})$ be its rational cohomology algebra. An endomorphism is a homomorphism of this graded cohomology…

Linear subvariety conjecture. If $Y$ is $f^{-1}$-invariant or $f^{-1}$-periodic, then $Y$ is a linear subspace of $X$.

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

Meng–Zhang's conjecture. $X$ is a toric variety.

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

Zhao's Dixmier conjecture. The Lie algebras $\mathbb{W}_n$ satisfy the Dixmier conjecture for $n\geq 1$; equivalently, every nonzero endomorphism of $\mathbb{W}_n$ is an automorphi…

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…

Folklore conjecture. Any non-constant endomorphism $X\to X$ must be bijective.

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…

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…

Conjecture on totally periodic curves. Every totally periodic curve on $\mathbb{P}^3$ is linear.

Sato's conjecture. Then $X$ is toric.

Let $Y$ be a smooth projective rational surface, and let $g:Y\to Y$ be a non-isomorphic surjective endomorphism. Sato's conjecture. Then $Y$ is toric. Nakayama confirmed this conje…

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…