Almost complex manifolds with small Nijenhuis tensor

We study existence of complex structures on semidirect products g ⊕ρ v, where g is a real Lie algebra and ρ is a representation of g on v. Our first examples, the Euclidean algebra e(3) and the Poincaré algebra e(2, 1), carry complex structures obtained by deformation of a regular complex structure on sl (2, C). We also exhibit a complex structure on the Galilean algebra G(3, 1). We construct next a complex structure on g ⊕ρ v starting with one on g under certain compatibility assumptions on ρ. As an application of our results we obtain that there exists k ∈ {0, 1} such that (S1)k × E(n) admits a left invariant complex structure, where S1 is the circle and E(n) denotes the Euclidean group. We also prove that the Poincaré group P4k+3 has a natural left invariant complex structure. In case dim g = dim v, there is an adapted complex structure on g ⊕ρ v precisely when ρ determines a flat, torsion-free connection on g. If ρ is self-dual, g ⊕ρ v carries a natural symplectic structure as well. If, moreover, ρ comes from a metric connection then g ⊕ρ v possesses a pseudo-Kähler structure. We prove that the tangent bundle TG of a Lie group G carrying a flat torsion-free connection ∇ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial\bar{\partial}$-closed, i.e., to strong Kahler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kahler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if $M$ is a nilmanifold (not a torus) endowed with an invariant complex structure $J$, then $(M,J)$ does not admit any symplectic form taming $J$. Moreover, we show that if a nilmanifold $M$ endowed with an invariant complex structure $J$ admits an SKT metric, then $M$ is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

An integral condition involving [formula omitted]-harmonic (0,1)-forms

The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space [Formula: see text], but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a complex structure on [Formula: see text]. When the latter is a complex-analytic manifold admitting just one complex structure, there is a unique quantization whose classical limit is [Formula: see text]. Then the notion of coherence is the same for all observers. However, when [Formula: see text] admits two or more nonbiholomorphic complex structures, there is one different quantization per different complex structure on [Formula: see text]. The lack of analyticity in transforming between nonbiholomorphic complex structures can be interpreted as the loss of quantum-mechanical coherence under the corresponding transformation. Observers using one complex structure perceive as coherent the states that other observers, using a different complex structure, do not perceive as such. This is the notion of a quantum-mechanical duality transformation: the relativity of the notion of a quantum.

Let L ¯ i →X i be a holomorphic line bundle over a compact complex manifold for i=1,2. Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S=S 1 ×S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant (ℂ * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 (X 1 ;ℝ)=0 and c 1 (L ¯ 1 )∈H 2 (X 1 ;ℂ) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for H q (S;𝒪 S ) when X i are projective manifolds, L ¯ i ∨ are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i ∨ are Cohen-Macaulay. We obtain applications to the Picard group of S. When X i =G i /P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over ℂ.

Among the known examples of Einstein manifolds, a good many are Kahler. In fact, all compact examples with zero or negative scalar curvature are either Kahler, or locally homogeneous. On a complex manifold, one often gets Kahler-Einstein metrics by specific techniques. One reason is perhaps, in the Kahler case, the relative autonomy of the Ricci tensor with regard to the metric, once the complex structure is given. The Ricci tensor—or, to be precise, the Ricci form—only depends on the volume form. On the other hand, on a compact manifold, the cohomology class of the Ricci form is determined by the complex structure (so that, in particular, the sign of an eventual Kahler-Einstein metric is itself determined by the complex structure). Due to these circumstances, it has been possible to exhibit some existence theorems of Einstein metrics in the Kahler framework (Calabi-Yau and Aubin-Calabi-Yau theorems) which have no counterpart in general Riemannian geometry.

Holomorphic geometry of non-perturbative superstring theories

In this chapter we shall introduce the very rich geometry of complex and Kähler manifolds. Complex manifolds are manifolds with a geometric structure called a complex structure, which gives every tangent space the structure of a complex vector space. They are defined in §4.1, together with complex submanifolds and holomorphic maps. Section 4.2 discusses tensors on complex manifolds, and their decomposition into components using the complex structure, and §4.3 defines holomorphic vector bundles over a complex manifold.

In this chapter we shall introduce the very rich geometry of complex and Kähler manifolds. Complex manifolds are manifolds with a geometric structure called a complex structure, which gives every tangent space the structure of a complex vector space. They are defined in 5.1, together with complex submanifolds and holomorphic maps. Section 5.2 discusses tensors on complex manifolds, and their decomposition into components using the complex structure, and 5.3 defines holomorphic vector bundles over a complex manifold.

A nonperturbative geometric formulation of the N=2 Neveu–Schwarz superstring theory which has been recently interpreted by H. Ooguri and C. Vafa [Mod. Phys. Lett. A5, 1389 (1990)] as a consistent quantum theory of self-dual gravity in four dimensions, is constructed. It is shown that the natural complex structure over the loop superspace ΩMd‖d associated to the N=2 Neveu–Schwarz fermionic string, is invariant under symmetry group OSp(2‖2)⊆SuperdiffS1‖2. Moreover, it is proved that there is a unique Lorentz and OSp(2‖2) invariant complex structure on ΩMd‖d. This result implies that the superspace of all admissible complex structures over ΩMd‖d is isomorphic to the homogeneous Kähler supermanifold SuperdiffS1‖2/OSp(2‖2). The Ricci curvature of SuperdiffS1‖2/OSp(2‖2) is calculated. Applying the method of geometric quantization to the N=2 Neveu–Schwarz superstring theory along the lines suggested by M. J. Bowick and S. G. Rajeev [Nucl. Phys. B361, 469 (1991)], a representation is constructed of nonperturbative N=2 superstring vacua in terms of antiholomorphic and horizontal sections of a certain vector bundle over SuperdiffS1‖2/OSp(2‖2); it is proved that such sections exist only in complex dimension d=2.

On the minimal sum of Betti numbers of an almost complex manifold

Given an almost complex structure J on a manifold M, a map f from a Riemann surface to M is called a pseudoholomorphic (or J-holomorphic) curve if at each point p of the surface, the ordinary differential is a complex linear map with respect to the complex structures j(p) and J(f(p)) on tangent spaces. In these lectures, we present the basic results on the theory of pseudoholomorphic curves in almost complex manifolds when the almost complex structure is tame by a symplectic form. Both local and global results are described. We then apply this theory to the construction of symplectic invariants of spaces and of diffeomorphisms of symplectic manifolds. We conclude with a brief description of the quantum cohomology of a symplectic manifold, which can be viewed as an application of the theory of pseudoholomorphic 2-spheres.

We review some properties of two special types of almost complex structures, introduced by T.‐J. Li and W. Zhang in [11], in relation to the existence of compatible symplectic structures and to the Hard Lefschetz condition. The two types of almost complex structures are defined respectively in terms of differential forms and currents. The paper is based on the results obtained in [9].We give a new example of an 8‐dimensional compact solvmanifold endowed with a C∞ pure and full almost complex structure calibrated by a symplectic form satisfying the Hard Lefschetz condition.

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally Kähler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-Kähler almost abelian Lie algebras.