By Daniele Angella
In those notes, we offer a precis of modern effects at the cohomological houses of compact complicated manifolds now not endowed with a Kähler structure.
On the single hand, the massive variety of built analytic ideas makes it attainable to end up robust cohomological homes for compact Kähler manifolds. at the different, in an effort to additional examine any of those homes, it's common to seem for manifolds that don't have any Kähler structure.
We concentration specifically on learning Bott-Chern and Aeppli cohomologies of compact advanced manifolds. numerous effects about the computations of Dolbeault and Bott-Chern cohomologies on nilmanifolds are summarized, permitting readers to check specific examples. Manifolds endowed with almost-complex buildings, or with different distinct constructions (such as, for instance, symplectic, generalized-complex, etc.), also are considered.
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Extra resources for Cohomological Aspects in Complex Non-Kähler Geometry
Let B be a complex (respectively, differentiable) manifold. A family fXt gt 2B of compact complex manifolds is said to be a complex-analytic (respectively, differentiable) family of compact complex manifolds if there exist a complex (respectively, differentiable) manifold X and a surjective holomorphic (respectively, 1 smooth) map W X ! t/ D Xt for any t 2 B, and (ii) is a proper holomorphic (respectively, smooth) submersion. A compact complex manifold X is said to be a deformation of a compact complex manifold Y if there exist a complex-analytic family fXt gt 2B of compact complex manifolds, and b0 ; b1 2 B such that Xb0 D Xs and Xb1 D Xt .
Brylinski, [Bry88, Sect. 2], is defined requiring that, for every k 2 N, and for every ˛; ˇ 2 ^k X , ˛ ^ ?! ˇ D ! n : nŠ By continuing in the parallelism between Riemannian geometry and symplectic geometry, one can introduce the d operator with respect to a symplectic structure ! 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, !
A ; dA / of dgas. B ; dB /, and a family Á Á C2j C1 ; dC2j C1 ◆◆◆ ◆◆◆ qqq q q ◆◆◆ q q q ◆◆' qis xqq qis Á C2j C2 ; dC2j C2 C2j ; dC2j of quasi-isomorphisms, varying j 2 f0; : : : ; n 1g. A ; dA / ; 0/. ^ X; d/ is a formal dga over R. A ; dA / be a dga over K. Given Œ˛12 2 H deg ˛12 A ; dA Œ˛34 2 H deg ˛34 ; Œ˛23 2 H deg ˛23 A ; dA ; and A ; dA such that Œ˛12 Œ˛23 D 0 let ˛13 2 Adeg ˛12 Cdeg ˛23 1 and Œ˛23 Œ˛34 D 0 ; and ˛24 2 Adeg ˛23 Cdeg ˛34 . 1/deg ˛12 ˛12 ˛23 D dA ˛13 and 1 be such that .