94 problems

Let $X$ be a minimal class VII surface with positive second Betti number $b_2(X)>0$. A global spherical shell is a smooth $3$-sphere with a neighborhood biholomorphic to a neighbor…

Tian–Jun Li's symplectic cone conjecture.

Global Spherical Shell Conjecture. Any non-Kählerian compact complex surface should belong to one of these classes.

Prasad–Yeung nonexistence conjecture. There are no fake projective planes in those remaining classes.

Class VII surface conjecture. All class VII surfaces with $b_2>0$ are Kato surfaces.

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

Let $X$ be a compact complex surface, and let $b_1(X)$ denote its first Betti number. Kodaira's conjecture. Every compact complex surface admits a Kähler structure if and only if ……

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…

For integers $p,q,p',q'$ defining nontrivial torus knots with $g(p,q)=g(p',q')$, let $F(p,q;p',q')$ be the fiber sum along a regular fiber of the rational surfaces $Z(p,q)$ and…

Let $S\subset \mathbb{P}^4$ be a smooth projective surface, and let $q(S)=h^1(S,\mathcal{O}_S)$ denote its irregularity. Van de Ven's conjecture. The irregularity satisfies … If tr…

A smooth Lefschetz fibration is a smooth fibration of an oriented 4-manifold over an oriented 2-manifold whose singular points have the local holomorphic form of two complex lines…

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…

Poincaré–Seiberg–Witten correspondence conjecture. If $p_g(V)=0$, then

The non-Kähler surface conjecture. Every non-Kähler complex surface is LCK.

Let $S$ be a surface of class VII$_0^+$, meaning that $b_1(S)=1$ and $b_2(S)>0$. A degeneration of blown-up primary Hopf surfaces is a surface arising as a degeneration of surfaces…

Let $S$ be a minimal compact complex surface of class VII$_0^+$, meaning that $b_1(S)=1$ and $n:=b_2(S)>0$, and let $\mathcal S\to B$ be its versal deformation over a ball of dimen…

Pinching-off decay conjecture. Either the Koiso metric or the Page metric decays to $\mathbb{CP}^2$ via a $\mathbb{CP}^1$ pinching off.

Let $S$ be a complex algebraic surface homeomorphic to … The preceding result states that all algebraic surfaces homeomorphic to $3\mathbb{C}P^{2}\#18\overline{\mathbb{C}P^{2}}$ ar…

Mandelbaum's ACD conjecture. Every simply connected complex algebraic surface is ACD. The paper notes that it is natural to extend this conjecture to simply connected symplectic fo…

Complex-structure conjecture for $Z(m,g)$. Although for the remaining values of $g$ there are complex manifolds in the homeomorphism type of $Z(m,g)$, no $Z(m,g)$ supports a comple…

Let $X$ be a complex surface, let $C$ be a holomorphically embedded submanifold, and let $Z(X,C,g)$ denote the manifolds constructed from this data. For finitely many values of $g$…

Let $S$ be a minimal compact complex surface of Kodaira class $VII_0$, with first Betti number $b_1(S)=1$, and suppose that $S$ contains more than $0$ rational curves, where…

Let $h_k$ be an analytic function on $|\operatorname{Im}(r)|\leq 1+\delta$ satisfying $h_k(-r)=h_k(r)$ and $|h_k(r)|\leq M[1+|\operatorname{Re}(r)|]^{-4-\delta}$, and define … For…

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…