Related papers: Almost complex blow-ups and positive closed $(1,1)…

We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-K\"ahler structure (J, \omega, g). In particular, almost-K\"ahler manifolds satisfying a…

In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ defined on the twistor space $J(M,g)$ of an even-dimensional manifold $M$ with additional structures $g$ and…

Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$.…

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 $[J, \Delta^m J]=0$. We prove a very…

In dimension 4, we extend the correspondence between compact nonsingular toric varieties and regular fans to a correspondence between almost complex torus manifolds and families of multi-fans in a geometric way, where an (almost) complex…

Reformulations of Donaldson's "tamed to compatible" question are obtained in terms of spaces of exact forms on a compact almost complex manifold $(M^{2n},J)$. In dimension 4, we show that $J$ admits a compatible symplectic form if and only…

The space $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ admits a natural homogeneous pseudo-Riemannian nearly Kaehler structure. We investigate almost complex surfaces in this space. In particular we obtain a complete classification of the…

We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for…

We prove the existence of torus invariant almost complex structure on any positively omnioriented four dimensional primitive quasitoric orbifold. We construct pseudo-holomorphic blowdown maps for such orbifolds. We prove a version of McKay…

This paper proves that on any tamed closed almost complex four-manifold $(M,J)$ whose dimension of $J$-anti-invariant cohomology is equal to the self-dual second Betti number minus one, there exists a new symplectic form compatible with the…

A hypercomplex manifold is by definition a smooth manifold equipped with two anticommuting integrable almost complex structures. For example, every hyperkaehler manifold is canonically hypercomplex (the converse is not true). For every…

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…

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension $2n\geq 10$…

We study Nakai-Moishezon type question and Donaldson's "tamed to compatible" question for almost complex structures on rational four manifolds. By extending Taubes' subvarieties--current--form technique to $J-$nef genus $0$ classes, we give…

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold…

We study compact almost complex manifolds admitting a Hermitian metric satisfying an integral condition involving $\overline \partial$-harmonic $(0,1)$-forms. We prove that this integral condition is automatically satisfied, if the…

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…

Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $$\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in \rho_{2} ( TH^{4}(M;\Z) ) \}.$$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset \rho_{2}(…

The paper is part of an attempt of understanding non-K\"ahler threefolds. We start by looking at compact complex non-K\"ahler threefolds with algebraic dimension two and admitting locally conformally K\"ahler metrics. Under certain…

Let $M$ be a real-analytic connected CR-hypersurface of CR-dimension $n>0$ having a point of Levi-nondegeneracy. The following alternative is demonstrated for both the symmetry algebra $s$ and the automorphism group $G$ of $M$. Denote by…