Invariant Complex Structures Research Articles

The classification of homogeneous Einstein metrics on flag manifolds with [formula omitted

Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

The Pluriclosed Flow on Nilmanifolds and Tamed Symplectic Forms

We study the evolution of strong Kahler with torsion (SKT) structures via the pluriclosed flow on complex nilmanifolds, i.e., on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces, we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow, and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving symplectic forms taming a complex structure, by considering the evolution of symplectic forms via the flow induced by the Bismut Ricci form.

Equi-harmonic maps and Plücker formulae for horizontal-holomorphic curves on flag manifolds

In this paper, we derive Plucker formulae for holomorphic maps into the maximal flag manifolds of the complex semi-simple Lie groups. Holomorphy is taken with respect to either an invariant complex structure or an invariant almost complex structure that takes part of a \((1,2)\)-symplectic Hermitian structure. The maps are assumed to be horizontal, in the case of a complex structure or to satisfy a generalization of this hypothesis in the \((1,2)\)-symplectic case. We also provide a relationship between holomorphic-horizontal curves and equiharmonic maps.

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.

Todd genera of complex torus manifolds

In this paper, we prove that the Todd genus of a compact complex manifold $X$ of complex dimension $n$ with vanishing odd degree cohomology is one if the automorphism group of $X$ contains a compact $n$-dimensional torus $\Tn$ as a subgroup. This implies that if a quasitoric manifold admits an invariant complex structure, then it is equivariantly homeomorphic to a compact smooth toric variety, which gives a negative answer to a problem posed by Buchstaber-Panov.

The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

In this paper we give a proof of the Gauss–Bonnet theorem of Connes and Tretkoff for noncommutative two tori \mathbb{T}_{\theta}^2 equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number \tau in the upper half plane, representing the conformal class of a metric on \mathbb{T}_{\theta}^2 , and a Weyl factor given by a positive invertible element k \in C^{\infty}(\mathbb{T}_{\theta}^2) , the value at the origin, \zeta (0) , of the spectral zeta function of the Laplacian \triangle\mkern-.5mu ' attached to (\mathbb{T}_{\theta}^2, \tau, k) is independent of \tau and k .

Invariant complex structures on tangent and cotangent lie groups of dimension six

This paper deals with left invariant complex structures on simply connected Lie groups, the Lie algebra of which is of the type Th D hE V, where is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on Th for h a three dimensional real Lie algebra. First it was proposed the study of complex structures J satisfying the constraint Jh D V. Whenever is the adjoint representation this kind of complex structures are associated to non-singular derivations of h. This fact allows different kinds of applications.

COMPLEX STRUCTURES ON FOUR-MANIFOLDS WITH SYMPLECTIC TWO-TORUS ACTIONS

We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a four-manifold with a symplectic two-torus action admits an invariant complex structure and give an identification of those that do not admit a Kähler structure with Kodaira's class of complex surfaces which admit a nowhere vanishing holomorphic (2,0)-form, but are not a torus nor a K3 surface.

A characterization of hyperbolic rational maps

In the early 1980's Thurston gave a topological characterization of rational maps whose critical points have finite iterated orbits (\cite{Th,DH1}): given a topological branched covering $F$ of the two sphere with finite critical orbits, if $F$ has no Thurston obstructions then $F$ possesses an invariant complex structure (up to isotopy), and is combinatorially equivalent to a rational map. We extend this theory to the setting of rational maps with infinite critical orbits, assuming a certain kind of hyperbolicity. Our study includes also holomorphic dynamical systems that arise as coverings over disconnected Riemann surfaces of finite type. The obstructions we encounter are similar to those of Thurston. We give concrete criteria for verifying whether or not such obstructions exist. Among many possible applications, these results can be used for example to construct holomorphic maps with prescribed dynamical properties; or to give a parameter description, both local and global, of bifurcations of complex dynamical systems.

Equitorsion holomorphically projective mappings of generalized Kählerian space of the first kind

In this paper we define generalized Kählerian spaces of the first kind ( $$ G_1^K N $$ ) given by (2.1)–(2.3). For them we consider hollomorphically projective mappings with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces ( $$ G_1^K N $$ and $$ G_1^{\bar K} N $$ ) and for them we find invariant geometric objects.

Uniqueness of the Fock quantization of a free scalar field onS1with time dependent mass

We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time dependent potential, also interpretable as a time dependent mass. Classically, the field satisfies a linear wave equation of the form $\ddot{\xi}-\xi"+f(t)\xi=0$. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field ($f=0$) provides a unitary implementation of the dynamics for sufficiently general mass terms, $f(t)$. Furthermore, this representation is uniquely specified, among the class of representations determined by $S^1$-invariant complex structures, as the only one allowing a unitary dynamics. These conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.

Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperk\ahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.

Uniqueness of the Fock representation of the Gowdy S1 × S2 and S3 models

After a suitable gauge fixing, the local gravitational degrees of freedom of the Gowdy S1 × S2 and S3 cosmologies are encoded in an axisymmetric field on the sphere S2. Recently, it has been shown that a standard field parametrization of these reduced models admits no Fock quantization with a unitary dynamics. This lack of unitarity is surpassed by a convenient redefinition of the field and the choice of an adequate complex structure. The result is a Fock quantization where both the dynamics and the SO(3)-symmetries of the field equations are unitarily implemented. The present work proves that this Fock representation is in fact unique inasmuch as, up to equivalence, there exists no other possible choice of SO(3)-invariant complex structure leading to a unitary implementation of the time evolution.

Complex structures, totally real and totally geodesic submanifolds of compact 3-symmetric spaces, and affine symmetric spaces

We construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group. Moreover, by making use of graded Lie algebras and some invariant structures of affine symmetric spaces, we classify half dimensional, totally real and totally geodesic submanifolds of a compact 3-symmetric space with respect to each invariant complex structure.

Hermitian structures on six-dimensional nilmanifolds

Let (J,g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kahler with torsion structures with nonabelian J on the nilmanifold \(\Gamma\backslash (H^3\times H^3),\) where H3 is the Heisenberg group. A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures (J,g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold \(\Gamma\backslash (H^3\times H^3)\) the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of \(H(2,1)\times \mathbb{R},\) where H(2,1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kahler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kahler metric.

STABILITY OF ABELIAN COMPLEX STRUCTURES

Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

Invariant Finsler metrics on homogeneous manifolds: II. Complex structures

In this paper, we study homogeneous complex Finsler spaces. We first prove that each homogeneous complex Finsler space can be written as a coset space of a Lie group with an invariant complex structure as well as an invariant complex Finsler metric. We then introduce the notion of Minkowski representations of Lie groups and Lie algebras to give a complete algebraic description for such spaces. Finally, we study symmetric complex Finsler spaces and obtain a complete classification of such spaces.

A GKM description of the equivariant cohomology ring of a homogeneous space

Let T be a torus of dimension n > 1 and M a compact T-manifold. M is a GKM manifold if the set of zero dimensional orbits in the orbit space M/T is zero dimensional and the set of one dimensional orbits in M/T is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about M is encoded in this graph. In this paper we prove that every compact homogeneous space M of non-zero Euler characteristic is of GKM type and show that the graph associated with M encodes geometric information about M as well as topological information. For example, from this graph one can detect whether M admits an invariant complex structure or an invariant almost complex structure.

DEFORMATION OF 2-STEP NILMANIFOLDS WITH ABELIAN COMPLEX STRUCTURES

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in three complex dimensions.

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C*-algebras.