34 problems

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.…

Let $X\subset\mathbb{C}^n$ and $Y\subset\mathbb{C}^m$ be complex analytic sets with $\dim X=\dim Y=d$. Their germs at zero are bi-Lipschitz homeomorphic if there is a bi-Lipschitz…

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

Yau's positivity conjecture. This invariant is positive for every normal surface singularity.

Analytic knot non-realizability conjecture. For any analytic structure $(Z,p)$ on $C(M)$, there exists an analytic knot in $M$ which is not realized by a holomorphic germ

Białynicki-Birula–Sommese conjecture. Part of the algebraic theory of GIT quotients should extend to the more general case of meromorphic actions of $({\mathbb C}^*)^k$ on a reduce…

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 $f:X\rightarrow Y$…

Relative Riemann–Hilbert correspondence. There is a one-to-one correspondence between pairs $(\mathcal{F},\nabla_f)$ consisting of an $\mathcal{O}_X$-coherent module and a flat tam…

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…

Let $P_1({\mathbf z},\varepsilon),\ldots,P_n({\mathbf z},\varepsilon):({\mathbb C}^{n+1},0)\to\mathbb C$ be holomorphic functions depending analytically on $\varepsilon$. For each…

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

A complex analytic variety is a space locally defined by complex analytic equations. A Whitney stratification is a decomposition into smooth strata satisfying Whitney's conditions;…

Openness conjecture. The subset $\mathfrak N$ is open. The source explains that this would follow from a minimal model program for projective morphisms between complex analytic spa…

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 $(X,x)$ and $(Y,y)$ be complex analytic germs, and let $F:(X,x)\to(Y,y)$ be a holomorphic map germ. A map is subflat when it satisfies Huckleberry's algebraic condition referre…

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.