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Cohomology of Almost-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

Let X be a differentiable manifold endowed with an almost-complex structure J. Note that if J is not integrable, then the Dolbeault cohomology is not defined. In this chapter, we are concerned with studying some subgroups of the de Rham cohomology related to the almost-complex structure: these subgroups have been introduced by T.-J. Li and W. Zhang in (Comm. Anal. Geom. 17(4):651–683, 2009), in order to study the relation between the compatible and the tamed symplectic cones on a compact almost-complex manifold, with the aim to throw light on a question by S.K. Donaldson, (Two-forms on four-manifolds and elliptic equations, Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 153–172, Question 2) (see Sect. 4.4.2), and it would be interesting to consider them as a sort of counterpart of the Dolbeault cohomology groups in the non-integrable (or at least in the non-Kähler) case, see Drǎghici et al. (Int. Math. Res. Not. IMRN 1:1–17, 2010, Lemma 2.15, Theorem 2.16). In particular, we are interested in studying when they let a splitting of the de Rham cohomology, and their relations with cones of metric structures.

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Notes

  1. 1.

    Recall that, given \(K \in \mathrm{End}(\mathit{TX})\) such that \({K}^{2} = \mathrm{id}_{\mathit{TX}}\), one can define, by duality, an endomorphism \(K \in \mathrm{End}({T}^{{\ast}}X)\) such that \({K}^{2} = \mathrm{id}_{{T}^{{\ast}}X}\), and hence one gets a natural decomposition \({T}^{{\ast}}X ={ \left ({T}^{+}X\right )}^{{\ast}}\oplus {\left ({T}^{-}X\right )}^{{\ast}}\) into eigen-bundles. Extending \(K \in \mathrm{End}({T}^{{\ast}}X)\) to \(K \in \mathrm{End}\left ({\wedge }^{\bullet }X\right )\), one gets a decomposition of the bundle of differential ℓ-forms, for \(\ell\in \mathbb{N}\): more precisely,

    $$\displaystyle{{\wedge }^{\ell}X\; =\;\bigoplus _{p+q=\ell} \wedge _{+\,-}^{p,\,q}\,X\qquad \text{ where }\qquad \wedge _{ +\,-}^{p,\,q}\,X\;:=\; {\wedge }^{p}{\left ({T}^{+}X\right )}^{{\ast}}\otimes {\wedge }^{q}{\left ({T}^{-}X\right )}^{{\ast}}\;;}$$

    If the almost-D-complex structure K is integrable, then the exterior differential splits as

    $$\displaystyle{\mathrm{d}\; =\; \partial _{+} + \partial _{-}}$$

    where

    $$\displaystyle{\partial _{+}:=\pi _{\wedge _{+\,-}^{p+1,\,q}\,X}\circ \mathrm{d}: \wedge _{+\,-}^{p,\,q}\,X \rightarrow \wedge _{ +\,-}^{p+1,\,q}\,X\quad \text{ and }\quad \partial _{ -}:=\pi _{\wedge _{+\,-}^{p,\,q+1}\,X}\circ \mathrm{d}: \wedge _{+\,-}^{p,\,q}\,X \rightarrow \wedge _{ +\,-}^{p,\,q+1}\,X}$$

    (where \(\pi _{\wedge _{+\,-}^{r,\,s}\,X}: \wedge _{+\,-}^{\bullet,\,\bullet }\,X \rightarrow \wedge _{+\,-}^{r,\,s}\,X\) denotes the natural projection onto ∧+ − r, s  X, for every \(r,s \in \mathbb{N}\)). In particular, the condition \({\mathrm{d}}^{2} = 0\) is rewritten as

    $$\displaystyle{\left \{\begin{array}{rcl} \partial _{+}^{2} & = & 0 \\ \partial _{+}\partial _{-} + \partial _{-}\partial _{+} & = & 0 \\ \partial _{-}^{2} & = & 0 \end{array} \right.\;,}$$

    and hence one can define a D-complex counterpart of the Dolbeault cohomology by considering the cohomology of the differential complex \(\left (\wedge _{+\,-}^{\bullet,\,q}\,X,\,\partial _{+}\right )\) for every \(q \in \mathbb{N}\), that is,

    $$\displaystyle{H_{\partial _{+}}^{\bullet,\bullet }(X; \mathbb{R})\;:=\; \frac{\ker \partial _{+}} {\mathrm{im}\partial _{+}}\;.}$$
  2. 2.

    For example, [AR12, p. 533], consider two compact manifolds X + and X − having the same dimension and consider the natural D-complex structure on X + × X −, i.e., the D-complex structure given by the decomposition \(T\left ({X}^{+} \times {X}^{-}\right ) ={ \mathit{TX}}^{+}\, \oplus \,{\mathit{TX}}^{-}\), where \(K\lfloor _{{\mathit{TX}}^{\pm }} = \pm \mathrm{id}_{T\left ({X}^{+}\times {X}^{-}\right )}\), for ± ∈ { +, −} (recall that every D-complex manifold is locally of this form, see, e.g., [CMMS04, Proposition 2]); one has that the vector space \(H_{\partial _{+}}^{0,0}\left ({X}^{+} \times {X}^{-}\right ) \simeq {\mathcal{C}}^{\infty }\left ({X}^{-}\right )\) does not have finite dimension.

  3. 3.

    See [MT11, Ros12a] for more results concerning deformations of (almost-)D-complex structures.

  4. 4.

    For an analogous result in the setting of almost-D-complex structures in the sense of F. R. Harvey and H. B. Lawson, see [AR12, Proposition 1.4]; for an analogous result in the setting of symplectic structures, see [AT12b, Proposition 2.4].

  5. 5.

    For an analogous result in the setting of almost-D-complex structures in the sense of F.R. Harvey and H.B. Lawson, see [AR12, Lemma 2.3]; for an analogous result in the setting of symplectic structures, see [AT12b, Lemma 3.2].

  6. 6.

    For analogous results in the context of almost-D-complex structures in the sense of F. R. Harvey and H. B. Lawson, see [AR12, Proposition 2.4]; for analogous results in the context of symplectic structures, see [AT12b, Proposition 3.3]; compare also with [FT10, Theorem 3.4], by A. M. Fino and A. Tomassini, for almost-complex structures.

  7. 7.

    I would like to thank Weiyi Zhang for having let me notice this fact.

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Angella, D. (2014). Cohomology of Almost-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_4

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