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Giovanni Bazzoni

Publications and source records attributed to Giovanni Bazzoni.

At least 19 recordsLinked to original sources
arXiv2025

Purely coclosed $\mathrm{G}_2$-structures on nilmanifolds -- II

This paper completes the classification of seven-dimensional nilpotent Lie groups endowed with a left-invariant purely coclosed $\text{G}_2$-structure, initiated by the first-named author and collaborators. In this previous work, the authors provided the classification of decomposable seven-dimensional nilpotent Lie groups and of the indecomposable ones up to step $4$ of nilpotency. Here, we address the case of indecomposable $5$- and $6$-step nilpotent Lie groups.

math.DG↗
arXiv2025

Complex symplectic structures: deformations and cohomology

We show that complex symplectic structures need not be preserved under small deformations, and we find sufficient conditions for this to happen. We study various cohomologies of compact complex symplectic manifolds, obtaining some topological obstructions to their existence.

math.DG↗
arXiv2023

Complex Symplectic Lie Algebras with Large Abelian Subalgebras

We present two constructions of complex symplectic structures on Lie algebras with large abelian ideals. In particular, we completely classify complex symplectic structures on almost abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)kähler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.

math.DG↗
arXiv2023

Moduli spaces of (co)closed $\mathrm{G}_2$-structures on nilmanifolds

We compute the dimensions of some moduli spaces of left-invariant closed and coclosed $\mathrm{G}_2$-structures on 7-dimensional nilmanifolds, showing that they are not related to the third Betti number. We also prove that, in contrast to the case of closed $\mathrm{G}_2$-structures, the group of automorphisms of a coclosed $\mathrm{G}_2$-structure is not necessarily abelian.

math.DG↗
arXiv2022

Spin-harmonic structures and nilmanifolds

We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.

math.DG↗
arXiv2021

Purely coclosed $G_2$-structures on nilmanifolds

We classify 7-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed $G_2$-structures. This is done by going through the list of all 7-dimensional nilpotent Lie algebras given by Gong, providing an example of a left-invariant 3-form $φ$ which is a pure coclosed $G_2$-structure (that is, it satisfies $d*φ=0$, $φ\wedge dφ=0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed $G_2$-structure for the rest of them.

math.DG↗
arXiv2021

Symmetric and skew-symmetric complex structures

On a complex manifold $(M,J)$, we interpret complex symplectic and pseudo-K\"ahler structures as symplectic forms with respect to which $J$ is, respectively, symmetric and skew-symmetric. We classify complex symplectic structures on 4-dimensional Lie algebras. We develop a method for constructing hypersymplectic structures from the above data. This allows us to obtain an example of a hypersymplectic structure on a 4-step nilmanifold.

math.DG↗
arXiv2018

Complex symplectic structures on Lie algebras

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplectic oxidation to classify eight-dimensional nilpotent complex symplectic Lie algebras.

math.SG↗
arXiv2018

Special types of locally conformal closed G$_2$-structures

Motivated by analogous results in locally conformal symplectic geometry, we study different classes of G$_2$-structures defined by a locally conformal closed 3-form. In particular, we give a complete characterization of invariant exact locally conformal closed G$_2$-structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G$_2$-structures.

math.DG↗
arXiv2018

Structure of locally conformally symplectic Lie algebras and solvmanifolds

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

math.DG↗
arXiv2017

Locally conformally symplectic and Kähler geometry

The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is "Locally conformal Kähler Geometry" by Sorin Dragomir and Liviu Ornea. Many progresses in this field, however, were accomplished after the publication of this book, hence are not contained there. On the other hand, there is no book on locally conformally symplectic geometry and many recent advances lie scattered in the literature. Sections 2 and 4 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.

math.DG↗
arXiv2017

On the history of the Hopf problem

This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on $S^6$?" This unsolved mathematical question was the subject of the Conference "MAM 1 $-$ (Non-)Existence of Complex Structures on $S^6$", which took place at Philipps-Universität Marburg, Germany, between March 27th and March 30th, 2017.

math.HO↗
arXiv2016

Homotopic properties of Kähler orbifolds

We prove the formality and the evenness of odd-degree Betti numbers for compact Kähler orbifolds, by adapting the classical proofs for Kähler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kähler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$.

math.DG↗
arXiv2016

Hereditary properties of co-Kähler manifolds

We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: $\dim(H^*(M;\mathbb{Q})) \geq 2^r$, for any $r$-torus $T^r$ which acts almost freely on $M$.

math.AT↗
arXiv2016

Parallel forms, co-Kähler Manifolds and their Models

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context.

math.DG↗