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Beatrice Brienza

Publications and source records attributed to Beatrice Brienza.

9 recordsLinked to original sources
arXiv2026

Hermitian curvature flow and HKT geometry

We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Amp\`ere equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

math.DG↗
arXiv2026

Sasaki with torsion manifolds and string backgrounds

Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-K\"ahler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-Einstein manifold in dimension $5$ and $7$. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with $S^1$.

math.DG↗
arXiv2025

The holonomy of the Obata connection on Joyce hypercomplex manifolds

We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $\mathrm{SU}(2n+1)$, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $\mathrm{SU}(2n+1)$ is more subtle: for every $n>1$, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $\mathrm{GL}(n(n+1),\mathbb{H})$. On the other hand, Soldatenkov showed that $\mathrm{SU}(3)$ has Obata holonomy equal to $\mathrm{GL}(2,\mathbb{H})$ \cite{Sol}, and we present here a new example on $\mathrm{SU}(5)$ with holonomy equal to $\mathrm{GL}(6,\mathbb{H})$. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lie in $\mathrm{SL}(n, \mathbb{H})$, yielding new compact examples of twisted Calabi-Yau manifolds.

math.DG↗
arXiv2025

Holonomy of the Obata connection on 2-step hypercomplex nilmanifolds

We study the holonomy of the Obata connection on 2-step hypercomplex nilmanifolds. By explicitly computing the curvature tensor, we determine the conditions under which the Obata connection is flat, showing that this depends on the nilpotency step of each complex structure. In particular, we show that for 2-step hypercomplex nilmanifolds the holonomy algebra of the Obata connection is always an abelian subalgebra of $\mathfrak{sl}(n, \mathbb{H})$ and we prove that the $\mathbb{H}$-solvable conjecture holds in this case. Furthermore, we provide new examples of $k$-step nilpotent hypercomplex nilmanifolds, with arbitrary $k$, which are not Obata flat.

math.DG↗
arXiv2025

On the structure of compact strong HKT manifolds

We study the geometry of compact strong HKT and, more generally, compact BHE manifolds. We prove that any compact BHE manifold with full holonomy must be K\"ahler and we establish a similar result for strong HKT manifolds. Additionally, we demonstrate a rigidity theorem for strong HKT structures on solvmanifolds and we completely classify those with parallel Bismut torsion. Finally, we introduce the Ricci foliation for hypercomplex manifolds and analyze its properties for compact, simply connected, 8-dimensional strong HKT manifolds, proving that they are always Hopf fibrations over a compact $4$-dimensional orbifold.

math.DG↗
arXiv2024

SKT solvable Lie algebras with codimension two nilradical

In the present paper we study SKT and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension two nilradical. We treat separately the case of $J$-invariant nilradical and non $J$-invariant nilradical. A classification of such SKT Lie algebras in dimension six is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds.

math.DG↗
arXiv2024

Generalized Kähler manifolds via mapping tori

Starting from the product of a $3$-torus and a compact Kähler (respectively, hyperKähler) manifold we construct via mapping tori generalized Kähler manifolds of split (respectively, non-split) type. In this way we obtain new non-Kähler examples and we recover the known examples of generalized Kähler solvmanifolds. Moreover, we investigate the formality and the Dolbeault cohomology of the generalized Kähler mapping tori.

math.DG↗
arXiv2024

CYT and SKT manifolds with parallel Bismut torsion

In the present paper, we study compact complex manifolds admitting a Hermitian metric which is SKT and CYT and whose Bismut torsion is parallel. We first obtain a characterization of the universal cover of such manifolds as a product of a Kaehler Ricci-flat manifold with a Bismut flat one. Then, using a mapping torus construction, we provide non-Bismut flat examples. The existence of generalized Kaehler structures is also investigated.

math.DG↗