12 problems

Let $G$ be a reductive algebraic group acting on affine space $\mathbb{A}^n$ over a field $k$. Linearization conjecture. The action is linear after a suitable polynomial change of…

Let $\Phi:\mathbb{C}^n\to\mathbb{C}^n$ be a polynomial automorphism of order two. The Linearization Conjecture states that there exists an automorphism…

Linearizable generation conjecture. The group generated by the linearizable automorphisms may equal the full automorphism group:

Let $X$ be a compact complex manifold of complex dimension $n$, and let $Y$ be a subvariety such that $X\setminus Y$ is biholomorphic to $\mathbb{C}^n$. The pair $(X,Y)$ is standar…

Linearization conjecture. Every such action should be conjugate to a linear action, equivalently, the acting subgroup should be conjugated into $\operatorname{GL}_n({\mathbb C})$.…

Varolin–Tóth conjecture. Then $M$ is biholomorphic to $\mathbb{C}^n$.

Let $\mathbf{T}=\mathbb{C}^{*d}$ act on affine space $\mathbb{C}^d$. Linearization conjecture. Any $\mathbf{T}$-action on $\mathbb{C}^d$ is linear after a suitable change of coordi…

Algebraic Linearization Conjecture. The group $G$ has a fixed point $p$, and the action of $G$ is linear with respect to a suitable coordinate system of $\mathbb{C}^n$ having $p$ a…

Let $k$ be a field of characteristic zero, let $A=k[X_1,\ldots,X_{n+1}]$, and let $f\in A$ satisfy … for every $\alpha\in\mathbb{C}$. A family of derivations is linearly independen…

Let $k$ be a field of characteristic zero and let $A=k[X_1,\ldots,X_{n+1}]$. For $D_1,\ldots,D_n\in\operatorname{LND}(A)$, write … A polynomial $f\in A$ is a coordinate if it is pa…

Shpilrain's generation conjecture.

Locally finite generation conjecture.