Related papers: Bott-Chern and $\bar\partial$ Harmonic forms on Al…

Let $(X,J)$ be a $4$-dimensional compact almost-complex manifold and let $g$ be a Hermitian metric on $(X,J)$. Denote by $\Delta_{\overline\partial}:=\overline\partial\overline\partial^*+\overline\partial^*\overline\partial$ the…

We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not…

Let $(M,J,g,\omega)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on $(M,J,g,\omega)$, proving that it is still elliptic. On a compact K\"ahler manifold, the kernels of the Dolbeault…

We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost K\"ahler manifolds $(M,J,\omega)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the…

We introduce an effective method to solve the $\bar\partial$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE…

Let $(X,J,\omega)$ be a compact $2n$-dimensional almost K\"ahler manifold. We prove primitive decompositions for Bott-Chern and Aeppli harmonic forms in special bidegrees and show that such bidegrees are optimal. We also show how the spaces…

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott-Chern and Aeppli cohomologies defined using the operators $d$, $d^c$. We explain how they are connected to already…

We study Dolbeault harmonic $(1,1)$-forms on compact quotients $M=\Gamma\backslash G$ of $4$-dimensional Lie groups $G$ admitting a left invariant almost Hermitian structure $(J,\omega)$. In this case, we prove that the space of Dolbeault…

Let $M$ be an $n$-dimensional $d$-bounded Stein manifold $M$, i.e., a complex $n$-dimensional manifold $M$ admitting a smooth strictly plurisubharmonic exhaustion $\rho$ and endowed with the K\"ahler metric whose fundamental form is…

We study the spaces of $(d + d^c)$-harmonic forms and $(d + d^\Lambda)$-harmonic forms, the natural generalization of the spaces of Bott-Chern harmonic forms, resp. symplectic harmonic forms from complex, resp. symplectic, manifolds to…

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension $h^{p,q}_{\bar \partial}$ of the kernel of the Dolbeault Laplacian independent of the choice of almost Hermitian…

Let $X=U/K$ be a compact Hermitian symmetric space, and let $\sE$ be a $U$-homogeneous Hermitian vector bundle on $X$. In a previous paper, we showed that the space of nearly holomorphic sections is well-adapted for harmonic analysis in…

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact K\"ahler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second…

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a…

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly…

In this note, we show that for a closed almost-K\"{a}hler manifold $(X,J)$ with the almost complex structure $J$ satisfies $\dim\ker P_{J}=b_{2}-1$ the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern…

Let $(M,J,g,\omega)$ be a K\"ahler manifold. We prove a $W^{1,2}$ weak Bott-Chern decomposition and a $W^{1,2}$ weak Dolbeault decomposition, following the $L^2$ weak Kodaira decomposition on Riemannian manifolds. Moreover, if the K\"ahler…

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, a special…

Let $x:M^m\to \bar M$, with $m\geq 3$, be an isometric immersion of a complete noncompact manifold $M$ in a complete simply-connected manifold $\bar M$ with sectional curvature satisfying $-c^2\leq K_{\bar M}\leq 0$, for some constant $c$.…

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as K\"ahler and hyperbolic…