arXiv subjects

Fabio Paradiso

Publications and source records attributed to Fabio Paradiso.

6 recordsLinked to original sources

arXiv2023

Hermitian structures on six-dimensional almost nilpotent solvmanifolds

We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally Kähler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-Kähler almost abelian Lie algebras.

math.DG↗

arXiv2022

Hermitian structures on a class of almost nilpotent solvmanifolds

In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients $Γ\backslash G$, where $G$ is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and $Γ$ is a lattice of $G$. We first obtain a characterization of Hermitian almost nilpotent Lie algebras $\mathfrak{g}$ whose nilradical $\mathfrak{n}$ has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In particular, we construct new examples of compact SKT manifolds. Finally, we prove some non-existence results for generalized Kähler structures in real dimension six. In higher dimension we construct the first examples of non-split generalized Kähler structures (i.e., such that the associated complex structures do not commute) on almost abelian Lie algebras. This leads to new compact (non-Kähler) manifolds admitting non-split generalized Kähler structures.

math.DG↗

arXiv2022

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedulli and L. Vezzoni and of the anomaly flow by D. H. Phong, S. Picard and X. Zhang on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.

math.DG↗

arXiv2021

Generalized Ricci flow on nilpotent Lie groups

We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

math.DG↗

arXiv2021

Locally conformally balanced metrics on almost abelian Lie algebras

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

math.DG↗

arXiv2021

Generalized Kähler almost abelian Lie groups

We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $\partial \bar \partial$-closed. We obtain a classification of six-dimensional generalized Kähler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generalized Kähler structure. Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow.

math.DG↗