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 over(∂, -) associated to the generalized complex structure. We prove that if the generalized ∂ over(∂, -)-lemma holds then the decomposition of forms gives rise to a decomposition of the cohomology of the manifold, H^• (M) = ⊕_{- n}ⁿ G H^k (M), and the canonical spectral sequence degenerates at E₁. We also show that if the generalized ∂ over(∂, -)-lemma holds, any generalized complex submanifold can be associated to a Poincaré dual cohomology class in the middle cohomology space G H⁰ (M).