Nijenhuis tensors in pseudoholomorphic curves neighborhoods

Nijenhuis tensors and obstructions for pseudoholomorphic mapping constructions

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 prolongations of the Cauchy-Riemann equation in 1-jets of the mapping from one almost complex manifold to another. We give a criterion of the prolongation existence of k-th pseudoholomorphic jets to the (k+1)-th ones. As a consequence basing on the Kuranishi's approach we obtain formal norm…

Tangent and normal bundles in almost complex geometry

In this paper we define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. These are intrinsically related to the Gromov D-operator. As an application we deduce normal forms of 1-jets of almost complex structures along a submanifold. In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the question of non-deformation and persistence of pseudoholomorphic curves.

On the symplectic realization of Poisson-Nijenhuis manifolds

We consider the problem of the symplectic realization of a Poisson-Nijenhuis manifold. By applying a new technique developed by M. Crainic and I. Marcut for the study of the above problem in the case of a Poisson manifold, we establish the existence, under a condition, of a nondegenerate Poisson-Nijenhuis structure on an open neighborhood of the zero-section of the cotangent bundle of the manifold, which symplectizes the initial structure. Additionally, we present some examples.

几乎微分非退化 Nijenhuis 算子

Almost Differentially Nondegenerate Nijenhuis Operators

针对特征多项式前 n−1 个系数函数独立、行列式任意的任意维 Nijenhuis 算子，本文给出其一般形式定理，并完整刻画行列式具非退化奇点的情形。

Taming the pseudoholomorphic beasts in $\mathbb{R}\times(S^1\times S^2)$

For a closed oriented smooth 4-manifold X with $b^2_+(X)>0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describes well-defined counts of pseudoholomorphic curves in the complement of the zero-set of such "near-symplectic" for…

Deformations of Noncompact Complex Curves and Meromorphic Envelopes of Spheres

We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex Kähler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite codimension analytic subset of a Banach ball, is given. The existence of this family is used to prove a generalization of Levi's continuity principle, which is applied to describe envelopes of meromorphy.