Orthogonal Complex Structures Research Articles

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $$\mathbb {F}=SU(3)/T^2$$ , and their configuration relative to the twistor projection $$\pi $$ from $$\mathbb {F}$$ to the complex projective plane $$\mathbb {P}^{2}$$ , defined with the help of an anti-holomorphic involution $$j$$ . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $$\mathbb {P}^3$$ of the 4-dimensional sphere $$S^4$$ . Deformations of twistor fibers project to real surfaces in $$\mathbb {P}^{2}$$ , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in $$\mathbb {F}$$ of bidegree $$(1,1)$$ . These surfaces define orthogonal complex structures on specified dense open subsets of $$\mathbb {P}^{2}$$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $$(1,1)$$ are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in $$\mathbb {F}$$ .

Slice regular functions and orthogonal complex structures over $\mathbb{R}^8$

This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the open mapping theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions.

Self-intersections of closed parametrized minimal surfaces in generic Riemannian manifolds

This article shows that for generic choice of Riemannian metric on a compact oriented manifold M of dimension four, the tangent planes at any self-intersection p in M of any prime closed parametrized minimal surface in M are not simultaneously complex for any orthogonal complex structure on M at p. This implies via geometric measure theory that H_2(M;{{mathbb {Z}}}) is generated by homology classes that are represented by oriented imbedded minimal surfaces.

Orthogonal bi-invariant complex structures on metric Lie algebras

This paper studies how many orthogonal bi-invariant complex structures exist on a metric Lie algebra over the real numbers. Recently, it was shown that irreducible Lie algebras which are additionally 2-step nilpotent admit at most one orthogonal bi-invariant complex structure up to sign. The main result generalizes this statement to metric Lie algebras with any number of irreducible factors and which are not necessarily 2-step nilpotent. It states that there are either 0 or $$2^k$$ such complex structures, with k the number of irreducible factors of the metric Lie algebra. The motivation for this problem comes from differential geometry, for instance to construct non-parallel Killing-Yano 2-forms on nilmanifolds or to describe the compact Chern-flat quasi-Kahler manifolds. The main tool we develop is the unique orthogonal decomposition into irreducible factors for metric Lie algebras with no non-trivial abelian factor. This is a generalization of a recent result which only deals with nilpotent Lie algebras over the real numbers. Not only do we apply this fact to describe the orthogonal bi-invariant complex structures on a given metric Lie algebra, but it also gives us a method to study different inner products on a given Lie algebra, computing the number of irreducible factors and orthogonal bi-invariant complex structures for varying inner products.

Killing Forms on 2-Step Nilmanifolds

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $k=2$ or $k=3$, we show that the space of Killing $k$-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

Majorana fermions and orthogonal complex structures

Ground states of quadratic Hamiltonians for fermionic systems can be characterized in terms of orthogonal complex structures. The standard way in which such Hamiltonians are diagonalized makes use of a certain “doubling” of the Hilbert space. In this work, we show that this redundancy in the Hilbert space can be completely lifted if the relevant orthogonal structure is taken into account. Such an approach allows for a treatment of Majorana fermions which is both physically and mathematically transparent. Furthermore, an explicit connection between orthogonal complex structures and the topological [Formula: see text]-invariant is given.

Orthogonal complex structures in

Orthogonal complex structures in domains in are studied using methods of multidimensional complex analysis. New results on removable singularities of such structures are established. The simplest multivalued orthogonal complex structures are investigated. A classification of quadrics in with respect to the action of the conformal group is given, and the discriminant sets of the twistor projections of model quadrics are described. Bibliography: 39 titles.

Non-existence of orthogonal complex structures on the round 6-sphere

In this short note, we review the well-known result that there is no orthogonal complex structure on S6 with respect to the round metric.

Non-existence of orthogonal complex structures on [formula omitted] with a metric close to the round one

I review several proofs for non-existence of orthogonal complex structures on the six-sphere, most notably by G. Bor and L. Hernández-Lamoneda, but also by K. Sekigawa and L. Vanhecke that we generalize for metrics close to the round one. Invited talk at MAM-1 workshop, 27–30 March 2017, Marburg.

The set of all orthogonal complex structures on the flat 6-tori

In [2], Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori TR2n for any n≥3. We will call these examples BSV-tori. In this note, we show that on a flat 6-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.

On a Complex-Symplectic Mirror Pair

We study the canonical Poisson structure on the loop space of the super-double-twisted-torus and its quantization. As a consequence we obtain a rigorous construction of mirror symmetry as an intertwiner of the N=2 super-conformal structures on the super-symmetric sigma-models on the Kodaira-Thurston nilmanifold and a gerby torus of complex dimension 2. As an application we are able to identify global moduli of equivariant generalized complex structures on these target spaces with moduli of equivariant orthogonal complex structures on the doubled geometry.

Twisted partially pure spinors

Motivated by the relationship between orthogonal complex structures and spure spinors, we define twisted partially pure spinors in order to characterize spinorially subspaces of Euclidean space endowed with a complex structure.

Twistor transforms of quaternionic functions and orthogonal complex structures

The theory of slice-regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega of \mathbb R^4 . When \Omega is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space \mathbb CP^3 .

Twistor geometry and warped product orthogonal complex structures

The twistor space of the sphere S2n is an isotropic Grassmannian that fibers over S2n. An orthogonal complex structure (OCS) on a subdomain of S2n (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this article, we use this correspondence to prove that any finite energy OCS on R6⊂S6 must be of a special warped product form, and we also prove that any OCS on R2n that is asymptotically constant must itself be constant. We give examples defined on R2n which have infinite energy and examples of nonstandard OCSs on flat tori in complex dimension 3 and greater.

Geometric properties of the magnetic Laplacian on the Euclidean 4-space

When the four-dimensional Euclidean space is endowed with a covariant derivative that is either self-dual or antiself-dual and of constant curvature, the corresponding magnetic Laplacian is closely related to the sub-Laplacian of the quaternionic Heisenberg group. Some geometric properties of this operator are studied. In particular, it is proved that there exists a canonical orthogonal complex structure which provides a factorization in the sense of Schrödinger.

A holonomy characterisation of Fefferman spaces

We prove that Fefferman spaces, associated to non-degenerate CR structures of hypersurface type, are characterised, up to local conformal isometry, by the existence of a parallel orthogonal complex structure on the standard tractor bundle. This condition can be equivalently expressed in terms of conformal holonomy. Extracting from this picture the essential consequences at the level of tensor bundles yields an improved, conformally invariant analogue of Sparling’s characterisation of Fefferman spaces.

Orthogonal complex structures on domains in $${\mathbb {R}^4}$$

An orthogonal complex structure on a domain in \({\mathbb {R}^4}\) is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in \({\mathbb {CP}^3}\) under the action of the conformal group SO°(1, 5). Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in \({\mathbb {R}^4}\) .

On a class of twistorial maps

We show that a natural class of twistorial maps gives a pattern for apparently different geometric maps, such as, ( 1 , 1 ) -geodesic immersions from ( 1 , 2 ) -symplectic almost Hermitian manifolds and pseudo horizontally conformal submersions with totally geodesic fibres for which the associated almost CR-structure is integrable. Along the way, we construct for each constant curvature Riemannian manifold ( M , g ) , of dimension m, a family of twistor spaces { Z r ( M ) } 1 ⩽ r < 1 2 m such that Z r ( M ) parametrizes naturally the set of pairs ( P , J ) , where P is a totally geodesic submanifold of ( M , g ) , of codimension 2 r, and J is an orthogonal complex structure on the normal bundle of P which is parallel with respect to the normal connection.

On the twistor space of pseudo-spheres

We give a proof that the sphere S 6 does not admit an integrable orthogonal complex structure using simple differential geometric methods. This appears as a corollary of a general analogous result concerning pseudo-spheres. We study the twistor space of a pseudo-Riemannian manifold in both the holomorphic and pseudo-Riemannian directions. In particular, we construct the twistor space of a pseudo-sphere S 2 q 2 n = SO 2 p + 1 , 2 q / SO 2 p , 2 q as a known pseudo-Kähler symmetric space. This leads to the explicit, unexpected computation of the exterior derivative of the Kähler form on the base manifold.

On a new construction of special Lagrangian immersions in complex Euclidean space

Special Lagrangian submanifolds of complex Euclidean space C have been studied widely over the last few years. These submanifolds are volume minimizing and, in particular, they are minimal submanifolds. When n = 2, special Lagrangian surfaces of C are exactly complex surfaces with respect to another orthogonal complex structure on R ≡ C. A very important problem here, and even a good starting point to study the not well understood problem about singularities of special Lagrangian n-folds in Calabi-Yau n-folds, is finding (non-trivial) examples of special Lagrangian submanifolds in C, since those are locally modeled on singularities of these. In [2], R. Harvey and H.B. Lawson constructed the first examples in C, where we point out the Lagrangian catenoid ([1, Remark 1], [3, Theorem A] and [5, Theorem 6.4]) to show a method of construction of special Lagrangian submanifolds of C using an (n-1)-dimensional oriented minimal Legendrian submanifold of S2n−1 and certain plane curves (for a better understanding, see Proposition A in section 2). Making use of this method, we include in section 2 three new examples of special Lagrangian submanifolds which have not been exposed yet. In [5], more examples of special Lagrangian submanifolds of C were constructed, all of them with cohomogeneity one, that is, the orbits of the symmetry group of the submanifold are of codimension one. Examples of special Lagrangian cones of cohomogeneity two were given in [3] and [5]. ∗Research partially supported by a MCYT and FEDER grant BFM2001-2967