arXiv subjects

Ettore Lo Giudice

Publications and source records attributed to Ettore Lo Giudice.

5 recordsLinked to original sources

arXiv2026

$p$-Kähler structures on nilmanifolds and holomorphically parallelizable manifolds

We prove the Alessandrini-Bassanelli conjecture on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds. As a consequence, we classify holomorphically parallelizable solvmanifolds admitting $p$-Kähler structures, for lower values of $p$. We further provide new examples of $(n-2)$-Kähler manifolds, in the setting of compact holomorphically parallelizable manifolds with reductive universal cover.

math.DG↗

arXiv2026

Holomorphically parallelizable solvmanifolds with special metrics and their deformations

We investigate the existence of strong Kähler with torsion metrics along deformations of the Iwasawa manifold and of the holomorphically parallelizable Nakamura manifold. We also show that the class of deformations of the holomorphically parallelizable Nakamura manifold yielding a non-left-invariant complex structure admits a balanced metric but does not admit any strong Kähler with torsion metric. We then construct the Kuranishi space of a $4$-dimensional holomorphically parallelizable solvmanifold and study whether small deformations of such a manifold admit SKT metrics. Finally, we provide some results concerning the existence of metrics satisfying $\partial \bar{\partial} ω= 0$, $\partial \bar{\partial} ω^2 = 0$ on a particular class of $2$-step nilpotent nilmanifolds.

math.DG↗

arXiv2025

Positive Hermitian curvature flow on 2-step nilpotent Lie groups

We study the positive Hermitian curvature flow for left-invariant metrics on $2$-step nilpotent Lie groups with a left-invariant complex structure $J$. We describe the long-time behavior of the flow under the assumption that $J[\mathfrak{g}, \mathfrak{g}]$ is contained in the center of $\mathfrak{g}$. We show that under our assumption the flow $g_{t}$ exists for all positive $t$ and $(G,(1+t)^{-1}g_{t})$ converges, in the Cheeger-Gromov topology, to a $2$-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups \cite{P2021, S2021}. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

math.DG↗

arXiv2025

p-Kähler structures on compact complex manifolds

Let $(M,J)$ be a complex manifold of complex dimension $n$. A $p$-Kähler structure on $(M,J)$ is a real, closed $(p,p)$-transverse form. In this paper, we address the conjecture of L. Alessandrini and G. Bassanelli on $(n-2)$-Kähler nilmanifolds equipped with nilpotent complex structures and holomorphically parallelizable nilmanifolds. We also derive necessary conditions for the existence of smooth curves of $p$-Kähler structures, starting from a fixed $p$-Kähler structure, along a differentiable family of compact complex manifolds. In addition, we study the cohomology classes of $p$-Kähler (resp. $p$-symplectic, $p$-pluriclosed) structures on compact complex manifolds. We provide several examples of families of compact complex manifolds admitting $p$-Kähler or $p$-symplectic structures.

math.DG↗

arXiv2024

$p$-symplectic and $p$-pluriclosed structures on solvmanifolds

Let $(M,J)$ be a $n$-dimensional complex manifold: a $p$-Kähler structure (resp. $p$-symplectic structure) on $M$ is a real, closed $(p,p)$-transverse form $Ω$ (resp. real, closed $2p$-form whose $(p,p)$-component is transverse). We give obstructions to the existence of such structures on compact complex manifolds. We provide several families of compact complex manifolds which admit both $(n-1)$-symplectic structures and special Hermitian metrics.

math.DG↗