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Giuseppe Barbaro

Publications and source records attributed to Giuseppe Barbaro.

14 recordsLinked to original sources
arXiv2026

On LCK Geometry of Gauduchon Connections

It is not a priori clear which of the Gauduchon connections is better suited to LCK and Vaisman manifolds. We thus investigate the geometry of these connections through Einstein problems and more general analytic and cohomological conditions on their Ricci tensors. Our results single out the Bismut connection as the privileged one for the non-K\"ahler lcK and Vaisman geometry. We therefore study the {second-Bismut--Einstein} equation and provide a characterization of these structures on Hopf manifolds.

math.DG↗
arXiv2026

On Bismut--Ambrose--Singer manifolds

We investigate Bismut--Ambrose--Singer (BAS) manifolds, namely Hermitian manifolds whose Bismut connection has parallel torsion and parallel curvature. We first establish a canonical reduction theorem for complete, simply-connected BAS manifolds. We then classify simply-connected BAS manifolds in the three fundamental homogeneous settings: the compact case, the non-compact semisimple case, and the nilpotent case. Building on this, we construct BAS manifolds in which these three geometries are combined, generalizing all previously known examples. Finally we classify complete, simply-connected, pluriclosed BAS manifolds.

math.DG↗
arXiv2026

Pluriclosed manifolds with parallel Bismut torsion

We present a complete classification of simply-connected pluriclosed manifolds with parallel Bismut torsion, extending previously known results in the literature. Consequently, we also establish a splitting theorem for compact manifolds that are both pluriclosed with parallel Bismut torsion and Calabi-Yau with torsion.

math.DG↗
arXiv2026

Rigidity results for non-Kähler Calabi-Yau geometries on threefolds

We derive a canonical symmetry reduction associated to a compact non-Kähler Bismut-Hermitian-Einstein manifold. In real dimension $6$, the transverse geometry is conformally Kähler, and we give a complete description in terms of a single scalar PDE for the underlying Kähler structure. In the case when the soliton potential is constant, we show that that the Bott-Chern number $h^{1,1}_{BC} \geq 2$, and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either $\SU(2) \times \mathbb R \times \mathbb C$ or $\SU(2) \times \SU(2)$.

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arXiv2025

Calabi-Yau locally conformally Kähler manifolds

We study compact locally conformally Kähler (lcK) manifolds which are Calabi--Yau, in the sense that $c_1^{BC}(X)=0$. First of all, we prove that all the known lcK manifolds which are Calabi--Yau are Vaisman. Then we prove that an lcK Chern--Ricci flat metric that is Gauduchon is necessarily Vaisman. Finally, specializing to Calabi--Yau solvmanifolds with left-invariant complex structure, we prove that a left-invariant metric is lcK if and only if it is Vaisman. Therefore, they are finite quotients of the Kodaira manifold.

math.DG↗
arXiv2025

Toric geometry of generalized Kähler-Ricci solitons

We establish a local equivalence between toric steady Kähler-Ricci solitons and $A$-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are $A$-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.

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arXiv2024

Bismut Hermitian Einstein metrics and the stability of the pluriclosed flow

We compute the (1,1)-Aeppli cohomology of compact simply-connected Lie groups. From this, we deduce that the Bismut flat metrics on the compact Bismut flat manifolds with finite fundamental group are globally stable for the pluriclosed flow. This prevents the existence of non-flat homogeneous Bismut Hermitian Einstein (hence also pluriclosed Calabi--Yau with torsion) metrics on C-spaces.

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arXiv2024

Generalized almost-Kähler-Ricci solitons

We generalize Kähler-Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map \cite{MR4017922}, and show that their existence is an obstruction to the existence of first-Chern-Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the $4$-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on $2n$-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler-Ricci solitons. In particular, we partially extend Matsushima's theorem \cite{MR0094478} to compact first-Chern-Einstein almost-Kähler manifolds.

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arXiv2023

A survey on rational curves on complex surfaces

In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the Kähler and non-Kähler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight the different approaches applied to the study of rational curves in each class. We also provide several examples and point out some open problems.

math.AG↗
arXiv2022

On the curvature of the Bismut connection: Bismut Yamabe problem and Calabi-Yau with torsion metrics

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis we construct explicit examples of Calabi--Yau with torsion Hermitian structures and prove a uniqueness result for them.

math.DG↗
arXiv2022

Global stability of the Pluriclosed flow on compact simply-connected simple Lie groups of rank two

We compute the (1,1)-Aeppli cohomology of compact simply-connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply-connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott-Chern and the Aeppli cohomologies, arise from just the left-invariant forms and we computed the whole Bott-Chern diamonds of SU(3) and Spin(5) when they are equipped with a left-invariant isotropic complex structure.

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arXiv2022

Second-Chern-Einstein metrics on 4-dimensional almost-Hermitian manifolds

We study 4-dimensional second-Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe 4-dimensional compact second-Chern-Einstein locally conformally symplectic manifolds and we give some examples of such manifolds. Finally, we study the second-Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second-Chern-Einstein metric with a parallel non-zero Lee form.

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arXiv2021

Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows

In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these examples we identify some HCFs which do not preserve Bismut-Griffiths-non-negativity.

math.DG↗