35 problems

Let $M$ be a fixed differentiable manifold underlying a K3 surface, and let … and $'$. Write $[]$ and $[']$ for their periods in ${\mathbb P}(H^2(M,{\mathbb C}))$, and let…

Let $M$ be a compact complex surface of general type, and let $\operatorname{Vol}_{K}(M)$ and $\operatorname{Vol}_{s}(M)$ denote its minimal volume defined using sectional curvatur…

Let $(M,J)$ be a compact complex surface, where $M$ is its underlying 4-manifold and $Y(M)$ is its Yamabe invariant. Negative Yamabe invariant conjecture. If … then $M$ is of gener…

Let $M$ be the underlying 4-manifold of a compact complex manifold of Kodaira dimension $0$ or $1$, and let $b_1(M)$ denote its first Betti number. Yamabe invariant conjecture for…

Let $M$ be the underlying 4-manifold of a complex algebraic surface of Kodaira dimension $\geq 0$. A sequence of unit-volume constant-scalar-curvature metrics on $M$ has uniformly…

Let $X_0$ be a nontrivial chain or ring of curves of genus greater than $0$, let $g$ be its arithmetic genus, and let $X_I$ be the family of curves over an interval described in th…

Let $M$ be a ruled compact complex surface, and let $\widetilde{M}$ be its blow-up at sufficiently many points. Blow-up conjecture. The surface $\widetilde{M}$ admits a scalar-flat…

Kähler–sG hyperbolicity conjecture.

Hamilton's closure symplectic cone conjecture.

Tian–Jun Li's symplectic cone conjecture.

Let $a4n$ act smoothly on $a4cp^2$. The action is homologically trivial if it acts trivially on the second homology $H_2(a4cp^2)$. A linear action is one induced by the standard li…

A compact complex surface in class VII$_0^+$ is a surface with first Betti number one and positive second Betti number. A Kato surface is a compact complex surface obtained by Kato…

Let $S$ be a compact complex surface in class $VII_0$, and let $\mathcal T(S)=]a,b[$, with $-\infty\leq a$ and $b<0$, be the moduli space of locally conformally symplectic structur…

Reid's conjecture. The fundamental group $\pi_1(X)$ is either finite or commensurable with the fundamental group of a curve.

Pointwise Lelong-number refinement. If $T$ is not in $L^2_{-1}(X)$ and $x otin L^2{-1}(T|X)$, then there exists on $X$ an exact positive $(1,1)$-current with non-vanishing Lelong n…

Lelong-number conjecture. If $X$ admits an exact positive $(1,1)$-current not in $L^2_{-1}(X)$, then there exists on $X$ an exact positive current with a non-vanishing Lelong numbe…

Hyperbolicity criterion. If all exact positive $(1,1)$-currents $T$ on $X$ are in $L^2_{-1}(X)$ and satisfy $I(T)=0$, then $X$ is hyperbolic.

Let $M$ be a compact LCK complex surface with proper potential, and let $\widetilde M$ be its Kähler $\mathbb Z$-covering. The metric-completion conjecture. The metric completion o…

Let $S$ be a compact complex surface in the Kodaira class ${\rm VII}$. Write ${\mathcal T}(S)$ for its set of Lee classes, viewed in $H^1_{dR}(S,\mathbb R)$ and ordered along…

Let $\operatorname{Bl}_{p}(\mathbb{C} \times \mathbb{P}^{1})$ be the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at a fixed point $p$ of the standard torus action, and let…

Let $M$ be any Inoue surface. Let $\omega(t)$ be the Chern-Ricci flow and let $\omega_\infty$ be the closed semipositive $(1,1)$-form representing a multiple of $c_1(K_M)$ describe…

Let $M^n$ be a compact complex manifold with nef canonical bundle $K_M$ and Kodaira dimension $\mathrm{Kod}(M)=n$. Let $\omega(t)$ solve the Chern-Ricci flow starting at an arbitra…

Rational-cycle conjecture. If every such $T$ belongs to $L^2_{-1}(X)$, but not every such $T$ satisfies $I(T)=0$, then $X$ admits a cycle of rational curves.

Global spherical shell conjecture. Every class $\mathrm{VII}_0$ surface $X$ with

Let $X$ be a smooth complex surface with $b_1(X)=0$ and $b_2(X)>1$. Suppose that $D_i$ are smooth, pairwise disjoint complex curves on $X$, each of positive genus $g_i>0$, and that…