2 problems

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The conjecture that all non-Kähler complex surfaces are locally conformally Kähler
The non-Kähler surface conjecture. Every non-Kähler complex surface is LCK.
Conjecturerefutedcomplex-geometry complex-surfaces locally-conformal-kahler-geometryarXiv posted 2 days ago
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Vanishing Bott–Chern class conjecture for LCK-manifolds embedded in Hopf manifolds
Vanishing Bott–Chern class conjecture. The Bott–Chern class of $M$ always vanishes.
Conjectureopenbott-chern-cohomology complex-geometry locally-conformal-kahler-geometryarXiv posted 2 days ago