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Cohomological Aspects in Complex Non-Kähler Geometry pp 1–63Cite as

Preliminaries on (Almost-)Complex Manifolds

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

Abstract

In this preliminary chapter, we summarize some basic notions and some classical results in (almost-)complex and symplectic geometry. In particular, we start by setting some definitions and notation concerning (almost-)complex structures, Sect. 1.1, symplectic structures, Sect. 1.2, and generalized complex structures, Sect. 1.3; then we recall the main results in the Hodge theory for Kähler manifolds, Sect. 1.4, and in the Kodaira, Spencer, Nirenberg, and Kuranishi theory of deformations of complex structures, Sect. 1.5; furthermore, we summarize some basic definitions and some useful facts about currents and de Rham homology, Sect. 1.6, and about nilmanifolds and solvmanifolds, Sect. 1.7, in order to set the notation for the following chapters. (As a matter of notation, unless otherwise stated, by “manifold” we mean “connected differentiable manifold”, and by “compact manifold” we mean “closed manifold”.)

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Notes

  1. 1.

    This fact can be generalized for a bi-differential bi-graded algebra of PD -type, [Kas12b, Sect. 2], see also [AK12, Sect. 2.1].

  2. 2.

    See, e.g., [Hum78, Sect. 7] for general results concerning \(\mathfrak{s}\mathfrak{l}(2; \mathbb{K})\)-representations.

  3. 3.

    Recall that an \(\mathfrak{s}\mathfrak{l}(2; \mathbb{R})\)-representation on a (possibly non-finite dimensional) \(\mathbb{R}\)-vector space V is called of finite H-spectrum if V can be decomposed into the direct sum of eigen-spaces of H and H has only finitely-many distinct eigen-values, [Yan96, Definition 2.2].

  4. 4.

    We recall that the Clifford algebra associated to \(\mathit{TX} \oplus {T}^{{\ast}}X\) and \(\left \langle \cdot \;\vert \;\cdot \cdot \right \rangle\) is

    $$\displaystyle{\mathrm{Cliff}\left (\mathit{TX} \oplus {T}^{{\ast}}X,\,\left \langle \cdot \,\vert \,\cdot \cdot \right \rangle \right )\,=\,\left.\left (\bigoplus _{ k\in \mathbb{Z}}\bigotimes _{j=1}^{k}\left (\mathit{TX} \oplus {T}^{{\ast}}X\right )\right )/\left \{v \otimes _{\mathbb{R}}v -\left \langle v\,\vert \,v\right \rangle: v \in \mathit{TX} \oplus {T}^{{\ast}}X\right \}\right..}$$
  5. 5.

    We recall that, given a manifold X endowed with a generalized complex structure \(\mathcal{J}\) , the canonical spectral sequence is the spectral sequence being naturally associated to the double complex obtained from \(\left (U_{\mathcal{J}}^{\bullet },\,\partial _{\mathcal{J},H},\,\overline{\partial }_{\mathcal{J},H}\right )\) : more precisely, to the double complex \(\left (U_{\mathcal{J}_{J}}^{\bullet _{1}-\bullet _{2}} \otimes _{\mathbb{C}}{\mathbb{C}\beta }^{\bullet _{2}},\,\partial _{\mathcal{J}_{ J}} \otimes _{\mathbb{R}}\mathrm{id},\,\overline{\partial }_{\mathcal{J}_{J}} \otimes _{\mathbb{R}}\beta \right )\) , where \(\left \{{\beta }^{m}\;:\; m \in \mathbb{Z}\right \}\) is an infinite cyclic multiplicative group generated by some β, [Cav05, Sect. 4.2], see [Bry88, Sect. 1.3], [Goo85, Sect. II.2], [Con85, Sect. II]; (compare also [AT13a, Sect. 1]).

  6. 6.

    We recall that a graded \(\mathbb{K}\)-algebra A • is graded-commutative if, for every x ∈ A degx and y ∈ A degy, it holds \(x \cdot y ={ \left (-1\right )}^{\deg x\cdot \deg y}\,y \cdot x\).

  7. 7.

    We recall that a differential \(\mathrm{d}: {A}^{\bullet } \rightarrow {A}^{\bullet +1}\) on a graded \(\mathbb{K}\)-algebra A • satisfies the graded-Leibniz rule if, for every x ∈ A degx and y ∈ A degy, it holds \(\mathrm{d}\left (x \cdot y\right ) = \mathrm{d}x \cdot y +{ \left (-1\right )}^{\deg x}\,x \cdot \mathrm{d}y\).

  8. 8.

    The author would like to thank Luis Ugarte for having pointed out the subject, and Junyan Cao and Gunnar Þór Magnússon for useful discussions on the matter.

  9. 9.

    Let C be a category with finite products, and consider Grp the category of groups; a group-object G in C is an object of C such that \(\mathrm{Hom}(\cdot,\,G): \mathrm{\mathbf{C}} \rightarrow \mathrm{\mathbf{Grp}}\) is a contravariant functor.

    In such a notation, a Lie group (respectively a Lie group of class \({\mathcal{C}}^{k}\), for \(k \in \mathbb{N} \cup \{\infty,\,\omega \}\), respectively a complex Lie group) is a group-object in the category of differentiable manifolds with differentiable maps (respectively in the category of manifolds of class \({\mathcal{C}}^{k}\) with maps of class \({\mathcal{C}}^{k}\), respectively in the category of complex manifolds with holomorphic maps). A homomorphism of Lie groups between G and H is a morphism f: G → H of manifolds inducing, for every manifold X, a homomorphism of groups between Hom(X, G) → Hom(X, H).

  10. 10.

    We recall that the following theorem by A.I. Mal’tsev: a group Γ is isomorphic to a discrete co-compact subgroup in a simply-connected connected nilpotent Lie group if and only if it is finitely-generated, nilpotent, and torsion-free, [Mal49, Corollary 1], see, e.g., [Rag72, Theorem 2.18].

  11. 11.

    Recall that a manifold is called parallelizable if its tangent bundle is trivial.

  12. 12.

    We recall the following, referring, e.g., to [TO97, Theorem 1.2]; see also [Rag72, Theorem 3.3, Corollary 3.5] for a different proof.

    Theorem 1.30 (Mostow Structure Theorem, [Mos54, Mos57, Theorem 2]).

    Any solvmanifold can be naturally fibred over a torus with a nilmanifold as a fibre. More precisely, let \(\left.\varGamma \setminus G\right.\) be a solvmanifold, and consider the maximal connected nilpotent subgroup N of G. Then (i) NΓ is a closed subgroup in G, (i) N ∩Γ is a lattice in N, and (i) \(\left.N\varGamma \setminus G\right.\) is a torus. Hence, one gets the Mostow bundle

    $$\displaystyle{\left.N \cap \varGamma \setminus N\right. = \left.\varGamma \setminus N\varGamma \right.\hookrightarrow \left.\varGamma \setminus G\right. \rightarrow \left.N\varGamma \setminus G\right.\;.}$$

    (See also [Boc09, Theorem 3.6], which gives a sufficient condition for the Mostow bundle to be a principal bundle.)

  13. 13.

    The notion of orbifold has been introduced by I. Satake in [Sat56], with the name of V-manifold, and has been studied, among others, by W.L. Baily, [Bai56, Bai54]. We recall that a complex orbifold of complex dimension n is a singular complex space whose singularities are locally isomorphic to quotient singularities \(\left.{\mathbb{C}}^{n}/G\right.\), for finite subgroups \(G\,\subset \,\mathrm{GL}(n; \mathbb{C})\), [Sat56, Definition 2]. We refer, e.g., to [Joy07, Joy00, Sat56, Bai56, Bai54] for more results concerning complex orbifolds and their cohomology.

  14. 14.

    Recall that nilmanifolds and solvmanifolds are Eilenberg and MacLane spaces of type K(π; 1); in particular, they are not simply-connected.

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Angella, D. (2014). Preliminaries on (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_1

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