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 ) = ⊕ − n n G H k ( M ) , and the canonical spectral sequence degenerates at E 1 . 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 G H 0 ( M ) .