arXiv research

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

Study on biharmonic almost complex structures on compact manifolds.

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

Almost complex structures found on many homotopy complex projective spaces.

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

Study functionals on almost complex structures for Yau's Challenge.

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

The paper studies lifts of complex structures on a manifold.

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…

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…

Study transverse Dolbeault cohomology for almost complex structures.

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

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

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…

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

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.…

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, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le α$, we prove that $\left(\sharp_{i=1}^α M_{i}\right) \sharp (α{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $α$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we…

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.

We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.

Improved optimal regularity for harmonic almost complex structures.

Study on Kodaira dimension of specific solvmanifolds without complex structures.

Study almost complex structures on six-manifolds using twistor spaces.

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…

New pseudo-Kähler Einstein spaces found with special almost complex structures.

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 …

The space ${\mathcal A}$ of almost complex structures on a closed manifold $M$ is studied. A natural parametrization of the space ${\mathcal A}$ is defined. It is shown, that ${\mathcal A}$ is a infinite dimensional complex weak Pseudo-Riemannian manifold. A curvature of the space ${\mathcal A}$ is found. The space ${\…

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…

Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.

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. …

Study of complex surfaces in a specific pseudo-Riemannian space.

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. As an application, we extend some known results concerning existen…

Computational techniques calculate dimensions of complex structures.

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is infinite dimensional, and also $0$- or $1$-dimensional. In the compact case, we …

On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a certain variational problem for almost complex structures compatible with the metric, for which the linearized Euler-Lagrange equation at Kähler-Einstein structures is given by the Dolbeault Laplacian acting on $(0,1)$-forms with values in the…

We prove that any quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if $M$ admits a positive omniorientation. In particular, we show that all obstructions to existence of $T^n$-invariant almost complex structure on $M^{2n}$ arise from cohomology of underlying polytope - and henc…

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

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

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

We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost co…

We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed alm…

New curvature equations obstruct integrability of complex structures.

In this paper, motivated by Chen--Ruan's stringy orbifold theory on almost complex orbifolds, we construct a new cohomology ring $\mathscr H^\ast_{G,cs}(X)$ for an equivariant almost complex pair $(X,G)$, where $X$ is a compact connected almost complex manifold, $G$ is a connected compact Lie group which acts on $X$ an…

Four-dimensional, oriented Lie algebras $\mathfrak{g}$ which satisfy the tame-compatible question of Donaldson for all almost complex structures $J$ on $\mathfrak{g}$ are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…

Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.

Overview of algebraic geometry for almost complex manifolds.

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

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…