The decomposition of forms and cohomology of generalized complex manifolds

We study the decomposition of forms induced by a generalized complex structure giving a complete description of the bundles involved and, around regular points, of the operators ∂∂ and ∂¯ associated to the generalized complex structure. We prove that if the generalized ∂∂¯-lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H•(M)=⊕−nnGHk(M), and the canonical spectral sequence degenerates at E1E1. We also show that if the generalized ∂∂¯-lemma holds, any generalized complex submanifold can be associated to a Poincaré dual cohomology class in the middle cohomology space GH0(M)GH0(M).