By Daniele Angella

In those notes, we offer a precis of modern effects at the cohomological homes of compact complicated manifolds no longer endowed with a Kähler structure.

On the only hand, the big variety of constructed analytic suggestions makes it attainable to turn out robust cohomological houses for compact Kähler manifolds. at the different, so that it will additional examine any of those houses, it's normal to seem for manifolds that don't have any Kähler structure.

We concentration particularly on learning Bott-Chern and Aeppli cohomologies of compact advanced manifolds. numerous effects in regards to the computations of Dolbeault and Bott-Chern cohomologies on nilmanifolds are summarized, permitting readers to check particular examples. Manifolds endowed with almost-complex constructions, or with different detailed constructions (such as, for instance, symplectic, generalized-complex, etc.), also are considered.

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**Example text**

Dc / g b^k X WD . 1/kC1 Á g d g for every k 2 N (note that, when J is integrable, then d D i @ @ ). 3], the pure-type components with respect to J of the harmonic representatives of the de Rham cohomology classes are themselves harmonic. 3]. 6. In the framework of generalized complex geometry, see Sect. 3, the d operator associated to a symplectic structure should be interpreted as the Á symplectic counterpart of the operator dc WD i @ @ associated to a complex structure, [Cav05]. Consider now, for k 2 N, the map Lk W ^n k X !

As d b^k X WD . 1/kC1 ?! d ?! for any k 2 N, and interpret it as the symplectic counterpart of the Riemannian d operator with respect to a Riemannian metric. -L. 7]. 1], [Kos85, p. 5. ; g/ on a compact manifold X (that is, ! TX/ is an almostcomplex structure on X , and g is a J -Hermitian metric on X such that ! -operator ?! and the Hodge- operator g are related by ?! dc / g b^k X WD . 1/kC1 Á g d g for every k 2 N (note that, when J is integrable, then d D i @ @ ). 3], the pure-type components with respect to J of the harmonic representatives of the de Rham cohomology classes are themselves harmonic.

UJ Define also, [Gua04a, p. 52], [Gua11, Remark at page 97], dJ H WD i @J ;H Á @J ;H W UJ ! 2 Cohomological Aspects of Generalized Complex Geometry Let X be a compact complex manifold endowed with an H -twisted generalized complex structure. L k By considering the decomposition ^ X ˝R C D k2Z UJ and the operators C1 C1 1 dH W UJ ! UJ ˚ UJ , and @J ;H W UJ ! UJ and @J ;H W UJ ! X I C/. X / < C1. J ;H By abuse of notation, one says that X satisfies the @J ;H @J ;H -Lemma if Á UJ ; @J ;H ; @J ;H satisfies the @J ;H @J ;H -Lemma, and one says that X satisfies J J the dH dJ H -Lemma if UJ ; dH ; dH satisfies the dH dH -Lemma.