arXiv research

The study characterizes real flag manifolds with invariant generalized almost complex structures.

Study transverse Dolbeault cohomology for almost complex structures.

Study on biharmonic almost complex structures on compact manifolds.

Study integrability of generalized almost complex structures on S^6.

Study of complex and Hermitian structures on specific Lie groups.

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

Study shows almost complex structures with certain tensor properties are prevalent.

Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.

In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongatio…

This article is mostly a writeup of two talks, the first given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It begins with a discussion of basic geometry of almost complex 6-manifolds. …

In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, Δ^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{cr…

Study integrability of specific geometric structures on odd Courant algebroids.

The paper studies lifts of complex structures on a manifold.

Generically an almost complex structure has no symmetries at all, but there exist symmetric structures. In this paper we describe how to guarantee that the pseudogroup of local symmetries is small (finite-dimensional). It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some …

Positivity of intersections in 4-manifolds leads to taming symplectic structures.

Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.

The paper classifies invariant structures on complex almost Abelian groups.

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the…

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

Study of complex structures on product twistor spaces for 4D manifolds.

Almost complex structures found on many homotopy complex projective spaces.

The paper connects complex contact structures to specific types of almost contact 3-structures.

Expanding on previous work, this note generalizes geometric structures results.

Based on recent work of T. Draghici, T.-J. Li and W. Zhang, we further investigate properties of the dimension h_J of the J-anti-invariant cohomology subgroup H_J of a closed almost Hermitian 4-manifold (M, g, J, F) using metric compatible and symplectic 2-form compatible almost complex structures. We prove that h_J = …

We define a generalized almost para-Hermitian structure to be a commuting pair $(\mathcal{F},\mathcal{J})$ of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähle…

To give an almost quaternionic structure on a 4n-manifold $M$ is equivalent to give its bundle of twistors $Z(Q)\longrightarrow M$. When $Q$ is invariant under a torsion free connection, $Z(Q)$ can be provided with an almost complex structure $\mathbb J$. In the case $n = 1$ Atiyah, Hitchin and Singer have related…

In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures.…

For the standard metric on the six-dimensional sphere, with Levi-Civita connection $\nabla$, we show there is no almost complex structure $J$ such that $\nabla_X J$ and $\nabla_{JX} J$ commute for every $X$, nor is there any integrable $J$ such that $\nabla_{JX} J = J \nabla_X J$ for every $X$. The latter statement gen…

Study on Kodaira dimension of specific solvmanifolds without complex structures.

Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.

The paper explores families of almost complex structures and transverse (p,p)-forms.

In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not solve the (currently still open) problem of determining whether there exists an int…

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

Overview of algebraic geometry for almost complex manifolds.

An \emph{$ω$-admissible almost complex structure} on a $2n$-dimensional symplectic manifold $(M,ω)$ is a $ω$-calibrated almost complex structure $J$ admitting a nowhere vanishing $\bar{\partial}_J$-closed $(n,0)$-form $ψ$. After giving some examples we consider the moduli space of admissible almost complex structures a…

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculate…

We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by dis…

Improved optimal regularity for harmonic almost complex structures.

A conformal change of $TM\oplus T^*M$ is a morphism of the form $(X,α)\mapsto(X,e^τα)$ $(X\in TM,α\in T^*M,τ\in C^\infty(M))$. We characterize the generalized almost complex and almost Hermitian structures that are locally conformal to integrable and to generalized Kähler structures, respectively, and give examples of …

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of $(\mathbb{R}^4,J)$, for some almost complex structure $J$ if and only if it is an elliptic curve. Furthermore we show that any (almost) complex $2n$-torus can be holomorphically embedded in $(\mathbb{R}^{4n},J)$ for a suitable almo…

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. …

In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimen…

Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

Unique complex structures on specific Lie algebras.