arXiv subjects

Giovanni Placini

Publications and source records attributed to Giovanni Placini.

14 recordsLinked to original sources

arXiv2026

Radial Projectively Induced Canonical K\"ahler Metrics: Rigidity and Classification

We study radial K\"ahler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a K\"ahler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini-Study metric, the flat metric, or, in complex dimension two, generalized Burns-Simanca metrics. We also prove that every radial projectively induced K\"ahler-Einstein metric has constant holomorphic sectional curvature and is therefore a Fubini-Study, flat, or complex hyperbolic metric. Finally, we show that a radial infinitely projectively induced extremal K\"ahler metric has unbounded maximal radial domain if and only if it is scalar-flat.

math.DG↗

arXiv2025

Strong formality of toric and homogeneous compact K\"ahler manifolds

All compact K\"ahler, or even $\partial\bar\partial$-manifolds, are rationally formal. Not all of them are strongly formal. Yet some of them are: For complete smooth complex toric varieties and homogeneous compact K\"ahler manifolds we show the stronger property that they are both rationally and strongly formal in a compatible way.

math.AT↗

arXiv2024

Approximation results for compact Vaisman manifolds

We extend the Tian approximation theorem for projective manifolds to a class of complex non-K\"ahler manifolds, the so-called Vaisman manifolds. More precisely, we study the problem of approximating compact regular, respectively quasi-regular, Vaisman metrics by metrics induced by immersions, respectively embeddings, into Hopf manifolds.

math.DG↗

arXiv2024

Nontrivial Massey products on compact Kähler manifolds

We show that the bigraded quasi-isomorphism type of the bigraded, bidifferential algebra of forms on a compact Kähler manifold generally contains more information than the de Rham cohomology algebra with its real Hodge structure. More precisely, on any closed Riemann surface of genus at least two, there is a nontrivial ABC-Massey product. Furthermore, starting from dimension three, there are simply connected projective manifolds with a nonzero ABC-Massey product of three divisor classes. In particular, compact Kähler manifolds are generally not formal in the sense of pluripotential homotopy theory.

math.AT↗

arXiv2023

Ricci iterations of well-behaved Kähler metrics

We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler--Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., $ρ_ω^k=λΩ$. In particular, when $k=1$, under some condition on the maximal domain of definition of canonical coordinates, we show that $λ$ is forced to be positive. Moreover, for arbitrary $k$, we prove two additional results. Namely, if $ω$ and $Ω$ are induced by a flat metric, then $ω$ is Ricci-flat. Finally, if a Kähler-Ricci soliton $Ω$ arises as Kähler--Ricci iteration of a metric $ω$ induced by a complex space form, then the Kähler--Ricci soliton is forced to be trivial, that is, Kähler--Einstein. These three theorems extend well known results on Kähler--Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.

math.DG↗

arXiv2023

Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, $η$-Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler-Ricci soliton on $\mathbb C^n$ which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds.

math.DG↗

arXiv2023

On weak and strict relatives K\"ahler manifolds

We study K\"ahler manifolds that are (weak) relatives, that is, K\"ahler manifolds which share a (locally isometric) submanifold. In particular, we prove that if two K\"ahler manifolds are weak relatives and one of them is projective, then they are relatives. Moreover, we introduce the notion of strict relatives K\"ahler manifolds and provide several nontrivial examples.

math.DG↗

arXiv2023

Approximation of Regular Sasakian Manifolds

We investigate the problem of approximating a regular Sasakian structure by CR immersions in a standard sphere. Namely, we show that this is always possible for compact Sasakian manifolds. Moreover, we prove an approximation result for non-compact $η$-Einstein manifolds via immersions in the infinite dimensional sphere. We complement this with several examples.

math.DG↗

arXiv2023

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi's rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

math.DG↗

arXiv2022

Any Sasakian structure is approximated by embeddings into spheres

We show that, for any given $q\geq 0$, any Sasakian structure on a closed manifold $M$ is approximated in the $C^{q}$-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold K\"ahler form by projectively induced ones given by Ross and Thomas in [21] in the $C^0$-norm to a $C^{q}$-approximation.

math.DG↗

arXiv2022

Maximally non-integrable almost complex structures: an $h$-principle and cohomological properties

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 $2n\geq 10$ (respectively $\geq 6$) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed $4$-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).

math.DG↗

arXiv2021

Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

Let $(g,X)$ be a Sasaki-Ricci soliton on a Sasakian manifold $S$. We prove that if $(S,g)$ admits a local Sasakian immersion in a Sasakian space form $S(N,c)$ of constant $ϕ$-sectional curvature $c$, then $S$ is $η$-Einstein and its $η$-Einstein constants are rational. Moreover, if $c\leq -3$, $S$ is locally equivalent to the Sasakian space form $S(n,c)$ and its $η$-Einstein constants are determined by $c$. Further results are obtained in the compact setting, i.e. when $c>-3$, under additional hypotheses.

math.DG↗

arXiv2019

Engel structures on complex surfaces

We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges. We also study associated Engel defining forms and define a unique splitting of $TM$ associated with $\mathcal{D}$ $J$-Engel.

math.DG↗

arXiv2015

Minimal symplectic atlases of Hermitian symmetric spaces

In this paper we compute the minimal number of Darboux chart needed to cover a Hermitian symmetric space of compact type in terms of the degree of their embeddings in $\mathbb{C} P^N$. The proof is based on the recent work of Y. B. Rudyak and F. Schlenk [18] and on the symplectic geometry tool developed by the first author in collaboration with A. Loi and F. Zuddas [12]. As application we compute this number for a large class of Hermitian symmetric spaces of compact type.

math.SG↗