Abstract
A construction is presented associating with any closed (1,1)-form ω on a complex manifold M a supermanifold, whose retract is the split supermanifold (M,Ω), where Ω is the sheaf of holomorphic forms on M. This supermanifold is non-split whenever ω is not ∂-exact. In particular, any complex line bundle L over M determines the supermanifold associated with the curvature form of a metric on L; it is non-split whenever the refined Chern class of L is non-zero. The case of a flag manifold M is studied in more details.
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Onishchik, A.L. A Construction of Non-Split Supermanifolds. Annals of Global Analysis and Geometry 16, 309–333 (1998). https://doi.org/10.1023/A:1006539601455
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DOI: https://doi.org/10.1023/A:1006539601455