Holomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic
Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology
is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore,
the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first
sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of
the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds
on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern
include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures
or parallelizable complex structures.
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