7 problems

Let $(M,g,J,\omega)$ be a compact Einstein almost-Kähler manifold. Goldberg's conjecture. The almost-Kähler structure is Kähler; equivalently, the almost complex structure $J$ is i…

Let $M$ be a compact $4$-manifold equipped with an almost complex structure $J$ and a taming symplectic form $\Omega$. Let $\sigma$ be a smooth volume form on $M$ satisfying … If…

The compact-case conjecture. Any compact, 4-dimensional almost Kähler manifold of pointwise constant totally-real sectional curvature is a self-dual Kähler surface.

Let $(M,J, omega)$ be a compact almost-Kähler manifold, where $J$ is an almost-complex structure and $omega$ is its fundamental $2$-form. Let $sigma$ be a smooth volume form on…

Let $M$ be a noncompact Einstein almost Kähler manifold, and suppose that every conformal vector field on $M$ is Killing. Rigidity Conjecture. Then $M$ must be isometric to complex…

The SCF maximal-existence conjecture. The solution $(\omega(t),g(t))$ exists smoothly as long as

Let $(M,g,J,\omega)$ be a closed almost Kähler manifold of real dimension $2n$. Define … and … If $F$ is defined by $F(\phi)\omega^n=(\omega(\phi))^n$, then Delanoë's result gives…