Cohomological decomposition of complex nilmanifolds

Abstract

Abstract. We study pureness and fullness of invariant complex structures on nilmanifolds. We prove that in dimension six, apart from the complex torus, there exist only two non-isomorphic complex structures satisfying both properties, which live on the real nilmanifold underlying the Iwasawa manifold. We also show that the product of two almost complex manifolds which are pure and full is not necessarily full.

On cohomological decomposition of generalized-complex structures

“…With respect to its natural holomorphically-parallelizable complex structure, the map [4,Theorem 3.1] and §3). The same holds when one endows the underlying differentiable manifold of I 3 with the Abelian complex structure given in §3 (see [27]). Such examples show that this kind of decomposition is a strictly weaker property than the ∂∂-Lemma.…”

“…As previously said, J 1 is complex-C ∞ -pure-and-full at every stage in the sense of Li and Zhang (see [26,Proposition 4] as regards to the first stage, see also [27]). In fact, one has…”

“…The complex structure gives rise to the generalized-complex structurewhere J * ∈ End(T * X) denotes the dual endomorphism of J ∈ End(T X).The Z-graduation on forms is given by [20, Example 4.25]Finally, note that ∂ = ∂ J and ∂ = ∂ J (see also [20, Remark 4.26]).Proof. Observe that in the almost-complex case the following equatily holdsIn general, the differential d of a generalized-complex structure can be decomposed asHowever, as J is actually an almost-complex structure one hasand therefore,Observe that these last equations yield to the following equivalent definition of the complex structure J 1 :J 1 e 1 = −e 2 , J 1 e 3 = e 4 , J 1 e 5 = e 6 .As previously said, J 1 is complex-C ∞ -pure-and-full at every stage in the sense of Li and Zhang (see [26, Proposition 4] as regards to the first stage, see also [27]). In fact, one has (e 1 , e 2 ) T 2 , Jwhere ω := −e 36 − e 45 and J : e 1 −→ e 2 .We note that, for every k ∈ Z, one hasON COHOMOLOGICAL DECOMPOSITION OF GENERALIZED-COMPLEX STRUCTURES 15 Therefore, we compute U 1 J = e 1 + i e 2 , U 0 J = 1, e 12 , U −1 J = e 1 − i e 2 , and U 2 ω = 1 − i e 36 − i e 45 − e 3456 , U 1 ω = e 3 − i e 345 , e 4 + i e 346 , e 5 + i e 356 , e 6 − i e 456 , U 0 ω = e 34 , e 35 , e 46 , e 56 , e 36 − e 45 , 1 + e 3456 , U −1 ω = e 3 + i e 345 , e 4 − i e 346 , e 5 − i e 356 , e 6 + i e 456 , U −2 ω = 1 + i e 36 + i e 45 − e 3456 .…”

“…In this case, it turns out that g 1,0 is actually an Abelian complex Lie algebra. From the general study accomplished in [27], one can conclude that there are only two left-invariant complex structures defined on M which are C ∞ -pure-and-full at every stage in the sense of Li and Zhang. They are precisely the holomorphicallyparallelizable stucture J 0 corresponding to the Iwasawa manifold (as proven in [4]) and the Abelian stucture J 1 in [1, Theorem 3.3].…”