arXiv research

The abstract discusses conditions for stable almost complex structures on specific 10-manifolds.

Study transverse Dolbeault cohomology for almost complex structures.

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

In order to look for a well-behaved counterpart to Dolbeault cohomology in D-complex geometry, we study the de Rham cohomology of an almost D-complex manifold and its subgroups made up of the classes admitting invariant, respectively anti-invariant, representatives with respect to the almost D-complex structure, miming…

We consider Courant and Courant-Jacobi brackets on the stable tangent bundle $TM\times\mathds{R}^h$ of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on $TM\times\mathds{R}^h$ can be prolonged to $TM\times\mathds{R}^k$,…

While small deformations of Kähler manifolds are Kähler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symp…

The Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stab…

We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed stable symplectic hypersurface V in a symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders (in addition to insertions on X\V) for stable V. We obtain invariants of the deformation cl…

In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the…

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', …

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invarian…

New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.

Study examines homology of contact CR-submanifolds in complex Euclidean space.

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

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

Study on biharmonic almost complex structures on compact manifolds.

The paper studies lifts of complex structures on a manifold.

The paper proves conditions for almost complex manifolds to admit almost complex structures through connected sums.

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

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

Almost complex structures found on many homotopy complex projective spaces.

New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.

We construct a family of compact almost Calabi--Yau manifolds of complex dimension 3 and therein a corresponding family of compact special Lagrangians with one-point singularities modelled upon that T^2-cone constructed by Harvey--Lawson and characterized by Haskins as a stable T^2-cone in the terminology by Joyce.

Stable generalized complex structures on certain surfaces are constant.

The paper proves smoothness of weakly biharmonic almost complex structures in dimension four.

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.…

The paper studies almost complex structures on ACH Einstein manifolds and finds deformation results.

Study on Kodaira dimension of specific solvmanifolds without complex structures.

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.

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

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…

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…

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.

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…

The paper examines complex and para-complex structures on pseudo-Riemannian spheres.

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

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

Study on energy-minimizing structures in complex geometry.

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.

This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the complex structures on an almost complex manifold. I also discuss the almost-Herm…

New method shows almost complex structures can be approximated smoothly.

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.

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.

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

Let $f$ be a Morse function on a closed manifold $M$, and $v$ be a Riemannian gradient of $f$ satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function $f$ associates to these data the Morse comple…