11 problems

Let $M$ be a compact locally conformally Kähler (LCK) manifold, and let $b_1(M)$ denote its first Betti number. Vaisman's conjecture. The number $b_1(M)$ should be odd. This conjec…

Let $(M,\theta,\omega)$ be an LCK manifold, let $d_\theta=d-\theta\wedge$ be its Morse–Novikov differential, and let $[\omega]\in H^2(M,L)$ be the Morse–Novikov class of…

Let $X$ be a compact locally conformally Kähler (lcK) manifold, and let $h^{1,2}(X)$ denote its Dolbeault number. A Hermitian metric is pluriclosed if its fundamental form is $^c$-…

A compact locally conformally Kähler (lcK) manifold is a compact complex manifold admitting a Hermitian metric whose fundamental form is locally conformal to a closed form. For a c…

Nonexistence conjecture. A compact, non-Kähler, locally conformally Kähler manifold cannot be endowed with various different special Kähler metrics, such as balanced, pluriclosed,…

Let $(M,I,g)$ be a compact Strominger Kähler-like (SKL) manifold of complex dimension at least $3$, where the Bismut connection is Kähler-like, and suppose that $g$ is not Kähler.…

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 a smooth bimeromorphic contraction be a contraction from a smooth compact complex manifold equipped with a locally conformally Kähler metric. In the particular case where an ex…

Let $M$ be a compact nilmanifold admitting a locally conformally Kähler (LCK) structure, and let $H(n)$ be the generalized Heisenberg group. Structure conjecture. Up to covering,…

Let $M$ be a compact locally conformally Kähler (LCK) manifold with automorphic potential, meaning that $M$ admits a Kähler covering with an automorphic potential. Monodromy conjec…

A locally conformally Kähler (LCK) manifold is a complex Hermitian manifold with Hermitian form $\omega$ satisfying $d\omega=\theta\wedge\omega$, where $\theta$ is a closed $1$-for…