By Daniele Angella
In those notes, we offer a precis of contemporary effects at the cohomological homes of compact advanced manifolds now not endowed with a Kähler structure.
On the single hand, the massive variety of constructed analytic thoughts makes it attainable to end up robust cohomological houses for compact Kähler manifolds. at the different, with a purpose to extra examine any of those homes, 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 complicated manifolds. a number of effects in regards to the computations of Dolbeault and Bott-Chern cohomologies on nilmanifolds are summarized, permitting readers to review 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 info for Cohomological Aspects in Complex Non-Kähler Geometry
TXI T X /. The i-eigenspace of the C-linear extension of J! T X ˝R C/ is LJ! D fX i ! /. The canonical bundle is UJn! 2], UJn! / Â Â exp Ã Ã ^ X ˝R C 2i : Note that, [Cav06, Sect. 2], dJ! D d : By considering the natural isomorphism 'W M Â Â ^ X ˝R C 3 ˛ ! / exp k k2Z 2i Ã Ã M ˛ 2 ^k X ˝R C ; k2Z one gets that, [Cav06, Corollary 1], ' ^ X ˝R C ' U n ; and ' d D @J! ' and ' dJ! D 2 i @J! ' I in particular, GH@ J! 3 (B-Transform, [Gua04a, Sect. 3]). Let X be a compact 2ndimensional manifold endowed with an H -twisted generalized complex structure J , and let B be a d-closed 2-form.
1, Corollary 2]). A manifold X endowed with an H -twisted generalized complex structure J satisfies J the dH dJ H -Lemma if and only if ker dH ; dH ,! UJ ; dH is a quasi-isomorphism of differential Z-graded C-vector spaces. In this case, it follows that the splitting L ^ X ˝R C D k2Z UJk gives rise to a decomposition in cohomology. 21] gives the following result. 1]). A manifold X endowed with an H -twisted generalized complex structure J satisfies the dH dJ Hlemma if and only if the canonical spectral sequence5 degenerates at the first level and the decomposition of complex forms into sub-bundles UJk , varying k 2 Z, induces a decomposition in cohomology.
5]. Furthermore, we note that A. Tomasiello proved in [Tom08, Sect. B] satisfying the d dJ -Lemma is a stable property under small deformations. 12]); see also [Hod35, Hod89]. 12], commutation relations on arbitrary Hermitian manifolds are provided; see also [Gri66], [Dem12, Sect. 1]). Let X be a compact Kähler manifold. Consider the differential operators @ and @ associated to the complex structure, the symplectic operators L and associated to the symplectic structure, and the Hodge- -operator associated to the Hermitian metric.
Cohomological Aspects in Complex Non-Kähler Geometry by Daniele Angella