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Cohomology of Complex Manifolds

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Cohomological Aspects in Complex Non-Kähler Geometry

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2095))

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Abstract

In this chapter, we study cohomological properties of compact complex manifolds. In particular, we are concerned with studying the Bott-Chern cohomology, which, in a sense, constitutes a bridge between the de Rham cohomology and the Dolbeault cohomology of a complex manifold.In Sect. 2.1, we recall some definitions and results on the Bott-Chern and Aeppli cohomologies, see, e.g., Schweitzer (Autour de la cohomologie de Bott-Chern, arXiv:0709.3528 [math.AG], 2007), and on the \(\partial \overline{\partial }\) -Lemma, referring to Deligne et al. (Invent. Math. 29(3):245–274, 1975). In Sect. 2.2, we provide an inequality à la Frölicher for the Bott-Chern cohomology, Theorem 2.13, which also allows to characterize the validity of the \(\partial \overline{\partial }\) -Lemma in terms of the dimensions of the Bott-Chern cohomology groups, Theorem 2.14; the proof of such inequality is based on two exact sequences, firstly considered by J. Varouchas in (Propriétés cohomologiques d’une classe de variétés analytiques complexes compactes, Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984, Lecture Notes in Math., vol. 1198, Springer, Berlin, 1986, pp. 233–243). Finally, in Appendix: Cohomological Properties of Generalized Complex Manifolds, we consider how to extend such results to the symplectic and generalized complex contexts.

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Notes

  1. 1.

    We recall that a proper holomorphic map f: X → Y from the complex manifold X to the complex manifold Y is called a modification if there exists a nowhere dense closed analytic subset B ⊂ Y such that \(f\lfloor _{X\setminus {f}^{-1}(B)}: X\setminus {f}^{-1}(B) \rightarrow Y \setminus B\) is a biholomorphism.

  2. 2.

    We get actually that, for every \(k \in \mathbb{N}\), it holds \(h_{\mathit{BC}}^{k} + h_{A}^{k} = 2\,h_{\overline{\partial }}^{k} + {a}^{k} + {f}^{k}\).

  3. 3.

    More in general, given a compact manifold X endowed with a Poisson bracket \(\left \{\cdot,\cdot \cdot \right \}\), and denoted by G the Poisson tensor associated to \(\left \{\cdot,\cdot \cdot \right \}\), by following J.-L. Koszul, [Kos85], one can define \(\delta:= \left [\iota _{G},\,\mathrm{d}\right ] \in {\mathrm{End}}^{-1}\left ({\wedge }^{\bullet }X\right )\). One has that δ 2 = 0 and \(\left [\mathrm{d},\delta \right ] = 0\), [Kos85, pp. 266, 265], see also [Bry88, Proposition 1.2.3, Theorem 1.3.1].

    It holds that, on any compact Poisson manifold, the first spectral sequence \({^\prime}E_{r}^{\bullet,\bullet }\) associated to the canonical double complex \(\left ({\mathrm{Doub}}^{\bullet,\bullet } {\wedge }^{\bullet }X,\,\mathrm{d} \otimes _{\mathbb{R}}\mathrm{id},\,\delta \otimes _{\mathbb{R}}\beta \right )\) degenerates at the first level, [FIdL98, Theorem 2.5].

    On the other hand, an example of a compact Poisson manifold (more precisely, of a nilmanifold endowed with a co-symplectic structure) such that the second spectral sequence \({^\prime}{^\prime}E_{r}^{\bullet,\bullet }\left ({\mathrm{Doub}}^{\bullet,\bullet } {\wedge }^{\bullet }X,\,\mathrm{d} \otimes _{\mathbb{R}}\mathrm{id},\,\delta \otimes _{\mathbb{R}}\beta \right )\) does not degenerate at the first level has been provided by M. Fernández, R. Ibáñez, and M. de León, [FIdL98, Theorem 5.1].

    In fact, on a compact 2n-dimensional manifold X endowed with a symplectic structure ω, the symplectic-⋆-operator \(\star _{\omega }: {\wedge }^{\bullet }X \rightarrow {\wedge }^{2n-\bullet }X\) induces the isomorphism \(\star _{\omega }: {^\prime}E_{r}^{\bullet _{1},\bullet _{2}}\stackrel{\simeq }{\rightarrow }{^\prime}{^\prime}E_{r}^{\bullet _{2},2n+\bullet _{1}}\), [FIdL98, Theorem 2.9].

  4. 4.

    We recall that the symplectic cohomologies, introduced by L.-S. Tseng and S.-T. Yau in [TY12a, Sect. 3], are defined as

    $$\displaystyle\begin{array}{rcl} H_{\mathrm{d}+{\mathrm{d}}^{\varLambda }}^{\bullet }(X; \mathbb{R})\;:=\; H_{\left (\mathrm{d},{\mathrm{d}}^{\varLambda };\mathrm{d}{\mathrm{d}}^{\varLambda }\right )}^{\bullet }({\wedge }^{\bullet }X)\;:=\; \frac{\ker \left (\mathrm{d} +{ \mathrm{d}}^{\varLambda }\right )} {\mathrm{im}\mathrm{d}{\mathrm{d}}^{\varLambda }} & & {}\\ \end{array}$$

    and

    $$\displaystyle\begin{array}{rcl} H_{\mathrm{d}{\mathrm{d}}^{\varLambda }}^{\bullet }(X; \mathbb{R})\;:=\; H_{\left (\mathrm{d}{\mathrm{d}}^{\varLambda };\mathrm{d},{\mathrm{d}}^{\varLambda }\right )}^{\bullet }\left ({\wedge }^{\bullet }X\right )\;:=\; \frac{\ker \mathrm{d}{\mathrm{d}}^{\varLambda }} {\mathrm{im}\mathrm{d} + \mathrm{im}{\mathrm{d}}^{\varLambda }}\;.& & {}\\ \end{array}$$

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  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Daniele Angella

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Angella, D. (2014). Cohomology of Complex Manifolds. In: Cohomological Aspects in Complex Non-Kähler Geometry. Lecture Notes in Mathematics, vol 2095. Springer, Cham. https://doi.org/10.1007/978-3-319-02441-7_2

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