21 problems

Let $f:({\mathbb C}^n,0)\rightarrow({\mathbb C},0)$ be a holomorphic function, with $n\ge 2$, and allow $f$ to have non-isolated singularities. The quantities $Q(f)$,…

Let $X$ be a Zariski open subset of an irreducible compact complex space. A compact irreducible complex space $Y$ is hyperbolic in the sense used by Noguchi, and $f:X\rightarrow Y$…

Let $f\colon (\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ be an injective holomorphic map germ. A map germ is corank $1$ when the differential $df_0$ has corank $1$. Lê's conjecture.…

Analytic MMP termination conjecture. After shrinking $Y$ around $W$, there exists a finite sequence of steps of a $(K_X+\Delta)$-minimal model program over $Y$ around…

Bobadilla–Kollár's conjecture. If $f$ is a $\mathbb Z$-homology fiber bundle, then $f$ is smooth.

Finite generation conjecture. The graded $\mathcal O_Y$-algebra $R(X/Y,K_X+\Delta)$ is locally finitely generated.

Neumann–Pichon's conjecture. $X$ is metrically conical at $0$ if and only if $X$ is locally metrically conical at $0$.

The general-type component conjecture. Then $g$ is Moishezon, and all other fibers are of general type over a possibly smaller disc.

The rational-singularity limit conjecture. Then $X_0$ is Moishezon.

The simple normal crossing degeneration conjecture. Under these assumptions, $g$ is Moishezon.

The local Moishezon morphism criterion. The morphism $g$ is locally Moishezon if and only if every irreducible component of every fiber $X_s$ is Moishezon.

Let $W_0\subset W$ be the base locus of the relevant smooth family, let $p^c\colon W_0\to D^c/\Gamma$ be the period map of the canonical moduli part, and set … Let $\mathcal M^c_W$…

Let $P$) be a sharp toric monoid, let $V_P$ be a neighborhood of the vertex $v$ of the associated toric space, and let $f\colon X\to Y=V_P$ be a smooth proper and exact morphism of…

Let $f:(\mathcal{U},\mathbf{0})\to(\mathbb{C},0)$ be a complex analytic function with one-dimensional critical locus at the origin, choose a generic linear coordinate $z_0$, and le…

Analytic structure conjecture. For every such $\mathcal O$, the unit $\mathcal O \to \overline{\Psi}(\mathcal O)^{\mathrm{alg}}$ has this universal analytification property. This g…

Transitivity conjecture. If $X \Subset_S Y$ and $Y \Subset_S Z$, then

Schapira's conjecture. If $\mathcal{M}$ is $d$-reflexive, then $j_*\mathcal{M}$ is a coherent $\mathcal{A}_X$-module.

Let $g: (\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be an injective map germ, and let $\operatorname{grad} g$ denote its gradient. The rank conjecture. The rank of the gradient at the ori…

Let $(0\in X)$ be an isolated singularity, and let $0\in U\subset X$ be a contractible, Stein neighborhood. Neighborhood-independence conjecture. The homeomorphism types of…

Let $X$ be a complex space, let $W$ be an irreducible component of $\operatorname{Arc}^{\circ}(X)$ or $\operatorname{ShArc}(X)$, and let $[\phi_1],\dots,[\phi_m]$ be arcs in $W$. C…

Let $X$ be a complex space, and let $\operatorname{Arc}^*(X)\subset \operatorname{Arc}(X)$ be the space of arcs that are not contained in $\operatorname{Sing} X$. A finite type fam…