Generalized holomorphic structures

The Atiyah class of generalized holomorphic vector bundles

“…For Gualtieri's definition, this is not true. In [26], the author gave the criterion for E to be equipped with a generalized complex structure in local coordinates, which turns out to be related with the Poisson structure (4) on the base manifold M . The intrinsic conditions for the total space to be a generalized complex manifold are still unknown.…”

“…Notice that the total space of a generalized holomorphic vector bundle is not necessarily a generalized complex manifold. The obstruction was discussed in [26]. We would like to restrict our attention to the category of generalized complex manifolds.…”

Rational Morita equivalence for holomorphic Poisson modules

“…Now, let W 0 := W − Sing(F σ ) ∪ Sing(W ) and consider φ 0 the restriction of φ on W 0 . In order to show that φ ∧ φ = 0 we follow the argument in [47] and next we extend φ 0 . Indeed, let θ be a local matrix representing the connection ∇ 0 .…”

Metric reduction and generalized holomorphic structures

“…A standard non-trivial example of generalized holomorphic bundles is the canonical line bundle in the pure spinor description of a generalized complex structure. Further analysis and examples of generalized holomorphic structures can be found in [13] [14].…”

“…The notion of generalized principal bundles was introduced in [14] to define generalized holomorphic structures in the setting of principal bundles. Definition 5.1.…”