17 problems

Conjecture 1.1.3. Under the curvature assumption given in Theorem, there exists a sufficiently large constant $\ell\geq e$, depending only on $\psi$ and $\delta$, such that for eve…

Extension property. Every non-degenerate holomorphic map

Let $(X,Y)$ be a smooth projective pair, with $X$ a surface and $Y$ a simple normal crossings divisor, and let $V_{\rm sing}$ be the singular locus of $Y$. Let $(L,h_L)$ be a line…

Let $X$ be a smooth projective variety, let $Y$ be a simple normal crossings divisor with singular locus $V_{\rm sing}$, let $(L,h_L)$ be a line bundle whose metric…

Giusto–Simpson conjecture. The following are equivalent over $\mathsf{RCA}_0$:

Let $X$ be a Kobayashi hyperbolic compact manifold of dimension $n$. Let ${\mathbb B}$ be the unit ball in $\mathbb C^p$, and let $E$ be a closed polar subset of ${\mathbb B}$. Nog…

Let $W$ be the hypersurface and let $\varphi$ be a plurisubharmonic weight for which the upper density $D^+_{\varphi}(W)$ is defined. The theorem asserting the relevant Bargmann–Fo…

Let $(X,\Delta)$ be a projective divisorial log terminal pair, with $\Delta$ a $\mathbb{Q}$-divisor, and write $\lfloor\Delta\rfloor=S$. Suppose that $K_X+\Delta$ is nef and … wher…

Let $f:M^n\rightarrow\mathbb{R}^{2n+p}$ be a real Kähler submanifold with rank $r>p$ everywhere. A Kähler extension of $f|_U$ is a real Kähler submanifold…

Let $X$ be a complex manifold, let $Y\subset X$ be a subvariety, and write … for the open embedding. Let $\mathcal{A}_X$ denote the relevant sheaf of microdifferential operators, a…

Let $E\subset\mathbb{R}^n$ be finite, let $N=\#(E)$, and write $\mathbb{X}=L^{m,p}(\mathbb{R}^n)$ with $n<p<\infty$. For subsets $S_1,\ldots,S_L\subset E$, denote by $f|_{S_\ell}$…

Let $(X,S+B)$ be an $n$-dimensional projective dlt pair such that $B$ is an effective $\mathbb Q$-divisor, and write $\lfloor S+B\rfloor=S$. Assume that $K_X+S+B$ is nef and … wher…

Let $(X,\Delta)$ be an $n$-dimensional projective divisorial log terminal pair such that $\Delta$ is a $\mathbb Q$-divisor, and write $\lfloor\Delta\rfloor=S$. Assume that…

Let $G \to \operatorname{GL}(n,\mathbb{C})$ be a representation with discrete kernel and closed image. Let $R$ be a torsion relation for $G$, and let $R'$ be the induced invariant…

Let $E \to M \to B$ be a holomorphic fibration whose total space is Kähler. A complex manifold is a Hartogs extension target for local biholomorphisms when local biholomorphisms in…

Let $M$ be a domain in a Stein manifold, let $\hat{M}$ be its envelope of holomorphy, and let a holomorphic first order structure on $M$ have an underlying holomorphic principal bu…

Let $M$ be a domain in a Stein manifold, let $\hat{M}$ be its envelope of holomorphy, and let $E \to M$ be a holomorphic principal bundle. The extension conjecture. The bundle $E$…