arXiv subjects
Publications and source records attributed to Riccardo Piovani.
We prove an $L^2$-$\partial\overline\partial$-Lemma involving smooth square integrable forms on complete K\"ahler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of $L^2$-forms has a gap in its spectrum near zero. This generalises the classical $\partial\overline\partial$-Lemma on compact K\"ahler manifolds.
We prove a Fr\"olicher inequality between $L^2$ Betti and $L^2$ Hodge numbers on normal coverings of compact complex manifolds. This is achieved by building an injection using suitable spectral projectors associated to the self-adjoint operators $(D_h)^2:=(\overline\partial+\overline\partial^*+h\partial+h\partial^*)^2$ for $h\in[0,1]$. With similar techniques, we show that the positivity of the spectrum of the Dolbeault Laplacian implies the positivity of the spectrum of the Hodge Laplacian; moreover, if equality holds in the $L^2$ Fr\"olicher inequality, then we can replace "positivity of the spectrum" with "spectral gap at 0" in the previous statement. As a by-product, in the case of compact complex manifolds, we find a new proof of the classical Fr\"olicher inequality which does not rely at all on spectral sequences and build an explicit injection from de Rham to Dolbeault cohomology.
Given a compact complex manifold, we study the cohomology and the Hodge theory for the elliptic complex of differential forms defined by Bigolin in 1969 and recently referred to as the Schweitzer complex. Recall that the double complex of a compact complex manifold decomposes into a direct sum of so-called squares and zigzags, and the zigzags are the only components contributing to cohomology. The main result of this paper states that in complex dimension 3, the multiplicities of zigzags in this decomposition are completely characterised by Betti, Hodge, Aeppli numbers plus Bigolin numbers, namely the dimensions of the Bigolin cohomology. The result is sharp, meaning that if we remove Hodge or Bigolin numbers from the previous statement then it becomes false. In addition, we compute the Bigolin cohomology on the small deformations of the complex structure of the Iwasawa manifold, and then apply the main theorem to fully describe the double complexes of all the small deformations. We also prove a Hodge decomposition for Bigolin harmonic forms on compact K\"ahler manifolds of any dimension. Finally, we partially extend the definition of this complex on almost complex manifolds, providing a cohomological invariant on $1$-forms which is finite when the manifold is compact.
We analyse the $L^2$ Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the $L^2$ Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.
We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira-Thurston manifolds.
Let $(M^{2n},J)$ be a compact almost complex manifold. The almost complex invariant $h^{p,q}_J$ is defined as the complex dimension of the cohomology space $\left\{\left[α\right]\in H^{p+q}_{dR}(M^{2n};\mathbb{C}) \,\vert\,α\in A^{p,q}(M^{2n}),\, dα= 0 \right\}$. When $2n=4$, it has many interesting properties. Endow $(M^{2n},J)$ with an almost Hermitian metric $g$. The number $h^{p,q}_d$, i.e., the complex dimension of the space of Hodge-de Rham harmonic $(p,q)$-forms, is almost Kähler invariant when $2n=4$. In this paper we study the relationship between $h^{p,q}_J$ and $h^{p,q}_d$ in dimension $2n\ge4$. We prove $h^{n,0}_J=0$ if $J$ is non integrable and show that $h^{p,0}_d$ is almost Kähler invariant. If $M^{2n}$ is a compact quotient of a completely solvable Lie group and $(J,g,ω)$ is left invariant, we find information also on $h^{1,1}_d$. Finally we study the $\mathcal{C}^\infty$-pure and $\mathcal{C}^\infty$-full properties of $J$ on $n$-forms for the special dimension $2n=4m$.
We study Dolbeault harmonic $(1,1)$-forms on compact quotients $M=Γ\backslash G$ of $4$-dimensional Lie groups $G$ admitting a left invariant almost Hermitian structure $(J,ω)$. In this case, we prove that the space of Dolbeault harmonic $(1,1)$-forms on $(M,J,ω)$ has dimension $b^-+1$ if and only if there exists a left invariant anti self dual $(1,1)$-form $γ$ on $(G,J)$ satisfying $id^cγ=dω$. Otherwise, its dimension is $b^-$. In this way, we answer to a question by Zhang.
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 locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.
We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost Kähler manifolds $(M,J,ω)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $ψ\in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $ω^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form $ψ\in \mathcal{H}_{D}^{k,k}$ are not $D$-harmonic, showing that the primitive decomposition of $(k,k)$-forms in general does not descend to harmonic forms.
Let $(M,J,g,ω)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on $(M,J,g,ω)$, proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott-Chern Laplacian coincide. We show that such a property does not hold when $(M,J,g,ω)$ is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira-Thurston manifold. Furthermore, if $(M,J,g,ω)$ is a connected compact almost Hermitian $4$-manifold, denoting by $h^{1,1}_{BC}$ the dimension of the space of Bott-Chern harmonic $(1,1)$-forms, we prove that either $h^{1,1}_{BC}=b^-$ or $h^{1,1}_{BC}=b^-+1$. In particular, if $g$ is almost Kähler, then $h^{1,1}_{BC}=b^-+1$, extending the result by Holt and Zhang for the kernel of Dolbeault Laplacian. We also show that the dimensions of the spaces of Bott-Chern and Dolbeault harmonic $(1,1)$-forms behave differently on almost complex 4-manifolds endowed with strictly locally conformally almost Kähler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott-Chern cohomology groups for almost complex manifolds, recently introduced by Coelho, Placini and Stelzig.
Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics $\{ω_t\}_t$ which equals a fixed SKT metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.
Let $(X,J,ω)$ be a compact $2n$-dimensional almost Kähler 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 of primitive Bott-Chern, Aeppli, Dolbeault and $\partial$-harmonic forms on $(X,J,ω)$ are related.
Let $(M,J,g,ω)$ be a Kähler 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 Kähler metric is complete and the sectional curvature is bounded, the $W^{1,2}$ Bott-Chern decomposition is strictly related to the space of $W^{1,2}$ Bott-Chern harmonic forms, i.e., $W^{1,2}$ smooth differential forms which are in the kernel of an elliptic differential operator of order $4$, called Bott-Chern Laplacian. We also generalize to the non compact case the well known property that on compact Kähler manifolds the kernel of the Dolbeault Laplacian and the kernel of the Bott-Chern Laplacian coincide.
Let $(M,J,g,ω)$ be a complete Hermitian manifold of complex dimension $n\ge2$. Let $1\le p\le n-1$ and assume that $ω^{n-p}$ is $(\partial+\overline{\partial})$-bounded. We prove that, if $ψ$ is an $L^2$ and $d$-closed $(p,0)$-form on $M$, then $ψ=0$. In particular, if $M$ is compact, we derive that if the Aeppli class of $ω^{n-p}$ vanishes, then $H^{p,0}_{BC}(M)=0$. As a special case, if $M$ admits a Gauduchon metric $ω$ such that the Aeppli class of $ω^{n-1}$ vanishes, then $H^{1,0}_{BC}(M)=0$.
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 $ρ$ and endowed with the Kähler metric whose fundamental form is $ω=i\partial\overline{\partial}ρ$, such that $i\overline{\partial}ρ$ has bounded $L^\infty$ norm. We prove a vanishing result for $W^{1,2}$ harmonic forms with respect to the Bott-Chern Laplacian on $M$.