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In this note, we determine the structure of the associative algebra generated by the differential operators μ¯,∂¯,∂\overline{\mu },\overline{\partial },\partial , and μ\mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [d,−]\left[d,-], as well as its cohomology with respect to all its inner differentials.

We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any (J , J N) holomorphic map from a compact pseudo-Hermitian manifold M with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold N with negative (resp. nonpositive) holomorphic sectional curvature is constant. We also construct explicit almost CR structures on a complex vector bundle over an almost CR manifold. [ABSTRACT FROM AUTHOR]

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of d-harmonic forms of pure bidegree. There is also a generalization of Hodge Index Theorem for compact almost Kähler 4-manifolds. In particular, these provide topological bounds on the dimension of the space of d-harmonic forms of pure bidegree, as well as several new obstructions to the existence of a symplectic form compatible with a given almost complex structure. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize the notion of almost analyticity in the almost (para)complex Lie algebroids framework. We use a global formalism which yields to generalizations of the main results of previous known almost (para)complex manifolds case. [ABSTRACT FROM AUTHOR]

We establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds. [ABSTRACT FROM AUTHOR]

We prove the existence of a maximal plurisubharmonic minorant of a given upper semicontinous function f on an almost complex manifold (M, J) of complex dimension two. Such maximal function is obtained as the pointwise minimum of averages offover the boundaries of all J-complex discs in M centered at the given point. This result is an almost complex analogue of classical results of Poletsky, Larusson and Sigurdsson, Rosay and others in the case when the almost complex structure J is integrable. [ABSTRACT FROM PUBLISHER]

The Runge approximation theorem for holomorphic maps is a fundamental result in complex analysis, and, consequently, many works have been devoted to extend it to other spaces ( e.g. maps between certain algebraic varieties or complex manifolds). This article presents such a result for pseudo-holomorphic maps from a compact Riemann surface to a compact almost-complex manifold M, given that the manifold M admits many pseudo-holomorphic maps from $${\mathbb {C}{\rm P}^1}$$ which can be thought of as local approximations of the Laurent expansion az + br/ z. This result specializes to some compact algebraic varieties ( e.g. rationally connected projective varieties). An application to Lefschetz fibrations is presented. [ABSTRACT FROM AUTHOR]

We study the deformation space of an embedded J-holomorphic disc D in an almost complex surface ( X, J). Every such J is shown to be equivalent to a small deformation of a certain model structure J β along D, where β : D→ℂ is a complex valued function whose modulus | β| is a biholomorphic invariant. Furthermore, we find a nonlinear invertible operator mapping the space of all small J-holomorphic deformations of the given J-holomorphic disc onto the space of small holomorphic deformations of the standard disc in ℂ2. [ABSTRACT FROM AUTHOR]