2 problems

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Torus-covering conjecture for GL(2,C)-structures
Torus-covering conjecture. Such manifolds are covered by compact complex tori.
Conjectureopencomplex-tori holomorphic-g-structures kahler-geometryarXiv posted 3 days ago
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Local homogeneity conjecture for holomorphic SL(2,C)-structures
Local homogeneity conjecture. Holomorphic $\operatorname{SL}(2,\mathbb C)$ -structures on compact complex manifolds of odd dimension are always locally homogeneous.
Conjectureopencomplex-geometry holomorphic-g-structures local-homogeneityarXiv posted 3 days ago

Torus-covering conjecture for GL(2,C)-structures