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Degeneracy of Holomorphic Poisson Spectral Sequence

Through the theory of Lie bi-algebroids and generalized complex structures, one could define a cohomology theory naturally associated to a holomorphic Poisson structure. It is known that it is the hypercohomology of a bi-complex such that one of the two operators is the classical $\overline{\partial}$-operator. Another operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The hypercohomology is naturally computed by one of the two associated spectral sequences. In a prior publication, the author of this article and his collaborators investigated the degeneracy of this spectral sequence on the second page. In this note, the author investigates the conditions for which this spectral sequence degenerates on the first page. Particular effort is devoted to nilmanifolds with abelian complex structures.