23 problems

Let $M$ be a compact $4$-manifold, let $J$ be an almost complex structure on $M$, and let $h^-_J$ denote the dimension of the real vector space of closed $J$-anti-invariant $2$-for…

Let $u$ be a weakly holomorphic and locally approximable map of the type considered in the paper, and let its singular set be measured by the bound referred to as the optimal size…

Dolbeault cohomology conjecture. The Dolbeault cohomology of $M$ coincides with its left-invariant Dolbeault cohomology:

Let $M$ be a compact $4$-manifold, let $J$ be an almost complex structure on $M$, and let $h^-_J$ denote the dimension of the real vector space of closed $J$-anti-invariant $2$-for…

Let $J$ be an almost complex structure on a manifold, and let $\alpha$ be a closed $J$-anti-invariant form, meaning that $J^*\alpha=-\alpha$. Draghici–Li–Zhang conjecture. The zero…

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a constraint manifold defined by an almost-complex structure tamed by $\Omega$…

Pali's potential conjecture. There exists a unique function $f\in Psh(X,J)\cap L^1_{loc}(X)$ whose associated distribution coincides with the distribution $u$.

Pali's conjecture. The function $u$ is plurisubharmonic if and only if the $(1,1)$-current

Let $(M^{2p},J)$ and $(N,J)$ be almost complex manifolds, and consider $J$-holomorphic maps in $W^{1,2}(M^{2p},N)$. Finite-measure conjecture. The singular set of such maps should…

Let $S^6$ carry its standard almost complex structure $J$, and let $\operatorname{Aut}(TS^6)$ denote the automorphisms of its tangent bundle. For $\psi\in\operatorname{Aut}(TS^6)$,…

Yau's conjecture. For $n\geq 3$, any closed $2n$-manifold admitting an almost complex structure will also admit a complex structure.

Let $(X,J)$ be a closed almost complex $4$-manifold. Suppose there exists a closed $2$-form $\phi$ such that, for every $x\in X$, there is a closed connected pseudoholomorphic curv…

Let a geometric local curve element and an equivariant local curve element be local curve elements of the same topological type in the framework of the paper. Geometric–equivariant…

Complex–almost complex comparison conjecture. This map is an isomorphism. The conjecture is intended to permit almost complex methods to study invariants of complex threefolds, but…

Generic polygon image conjecture. The map $\alpha: \mathsf{TS}_{k}\rightarrow\mathsf{TP}_{k+6}$ has image in $\mathsf{TP}^{\mathsf{gen}}_{k+6}$ and is a homeomorphism onto its imag…

Let $M$ be a closed complex manifold of real dimension $2n$, and let the total Betti number of $M$ mean the sum of all its Betti numbers. Sullivan's conjecture. The minimal total B…

Let $M$ be a compact, simply-connected, nonspin four-dimensional almost complex manifold. An energy-minimizing biharmonic almost complex structure on $M$ determines a homotopy clas…

Let $(g,J)$ be a compatible almost Hermitian structure, and define … A biharmonic almost complex structure is a zero of $B$. Transversality conjecture. The map $B$ is transverse to…

Let $M$ be a closed Kähler manifold with complex structure $J_0$. A positive almost complex structure is one satisfying the technical positivity condition introduced in the paper.…

Donald's conjecture. Yau's theorem can be extended to the case of general symplectic four-manifolds with compatible almost complex structures, at least when $b^+(M)=1$.

Epsilon-regularity conjecture. There exist constants $\varepsilon,C,r_0>0$, depending only on $\Omega$, $J$, and $\sigma$, such that if

Let $(M,\Omega)$ be a compact symplectic four-manifold with $b^+(M)=1$, equipped with an almost complex structure $J$ tamed by $\Omega$. Let $g_{\Omega}$ be the metric determined b…

Let $M$ be a minimal almost complex $4$-manifold, let $J$ be an almost complex structure on $M$ tamed by a symplectic form $\omega$, and let $r_{SW}$ denote the dimension of the Se…