4 problems

Gromov's rational inessentiality conjecture. Then $M$ is not rationally essential.

Gromov's rational essentialness conjecture. A closed $\mathbb{Q}$-essential $n$-manifold does not admit any Riemannian metric with positive scalar curvature.

Let $\mathcal{A}$, $\mathcal{HS}$, and $\mathcal{S}$ denote respectively the classes of algebraic manifolds, symplectic manifolds with the hard Lefschetz property, and symplectic m…

Essentialness conjecture. An essential manifold does not admit a metric of positive scalar curvature. The paper explains that essential spin$^c$ manifolds are K-theoretically essen…