arXiv subjects

Aleksandra Borówka

Publications and source records attributed to Aleksandra Borówka.

8 recordsLinked to original sources

arXiv2026

Adapted connections with skew-torsion on metric $f$-manifolds

We provide a natural higher-dimensional generalization of the adapted connections with skew-torsion on almost Hermitian and almost contact metric manifolds presented in \cite{FrIv}. We prove that a metric $f$-manifold $(M^{2n+s}, ϕ, ξ_i, η_j, g)$ with commuting characteristic vector fields admits a metric connection $\nabla$ with skew-torsion $T$ preserving the structure if and only if each Reeb vector field $ξ_i$ is Killing and the associated Nijenhuis tensor is totally skew-symmetric. This connection is uniquely determined by its torsion 3-form $T$, for which we derive its explicit formula. We further establish necessary and sufficient conditions for contact metric $f$-manifolds to admit such a connection. To this end, for $s\geq 2$ we construct a broad new class of geometries with parallel skew-torsion in terms of $\mathcal{S}$-manifolds, i.e., higher-dimensional analogues of Sasakian manifolds. We illustrate our results by presenting examples based on low-dimensional Lie groups.

math.DG↗

arXiv2026

Contact variation of almost contact pseudo-metric structures

For an almost contact pseudo-metric manifold we study a variation of the metric in the spirit of Meli-Ngakeu-Olea. Such variation is defined using a vector field on the manifold and the procedure shifts the signature of the metric by increasing the positive index by two. First we derive conditions on the vector field for the obtained manifold to remain almost contact pseudo-metric. Then we study the cases of almost pseudo co-Kähler manifolds. In particular we obtain geometric characterizations of variations for which the fundamental form remains closed and establish a local decomposition into a product. We further investigate the variation in the setting of statistical structures.

math.DG↗

arXiv2025

Complex quaternionic manifolds and c-projective structures

We discuss complex quaternionic manifolds, i.e., those that have holonomy $GL(n,\mathbb{H})U(1)$, which naturally arise via quaternionic Feix--Kaledin construction. We show that for a fixed c-projective class, any real analytic connection with type $(1,1)$ curvature induces, via quaternionic Feix--Kaledin construction, an $S^1$-invariant connection with holonomy contained in $GL(n,\mathbb{H})U(1)$. As an application, we characterize in this setting the distinguished $U^*(2n):=SL(n,\mathbb{H})U(1)$ connection studied in Battaglia \cite{Bat} and Hitchin \cite{Hit3}.

math.DG↗

arXiv2024

A note on dual abelian varieties

For any non-principal polarisation $D$, we explicitly construct $D$-polarised abelian variety $A$, such that its dual abelian variety is not (abstractly) isomorphic to $A$. For $\dim(A)>3$ the construction includes examples with submaximal Picard number equal to $(\dim(A)-1)^2+1$. As a corollary, we show that a very general non-principally polarised abelian variety is not isomorphic to its dual. Moreover, we show an example of an abelian variety that is isomorphic to its dual, yet it does not admit a principal polarisation.

math.AG↗

arXiv2019

Quaternion-Kähler manifolds near maximal fixed points sets of $S^1$-symmetries

Using quaternionic Feix--Kaledin construction we provide a local classification of quaternion-Kähler metrics with a rotating $S^1$-symmetry with the fixed point set submanifold $S$ of maximal possible dimension. For any Kähler manifold $S$ equipped with a line bundle with a unitary connection of curvature proportional to the Kähler form we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix-Kaledin construction from this data. Conversely, we show that quaternion-Kähler metrics with a rotating $S^1$-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when $S$ is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on $S$ by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix--Kaledin construction form $S$ are related by hyperkähler/quaternion-Kähler correspondence.

math.DG↗

arXiv2018

C-projective symmetries of submanifolds in quaternionic geometry

The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type $(1,1)$ curvature is a submanifold of a submaximally symmetric quaternionic model, and show how this fits into the construction. We give conditions for when the c-projective symmetries extend from the fixed points set of $v$ to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi-- and Eguchi-Hanson hyperkähler structures, showing that in some cases all quaternionic symmetries are obtained in this way.

math.DG↗

arXiv2017

$S^6$ and the geometry of nearly Kähler $6$-manifolds

We review results on and around the almost complex structure on $S^6$, both from a classical and a modern point of view. These notes have been prepared for the Workshop "(Non)-existence of complex structures on $S^6$" (\emph{Erste Marburger Arbeitsgemeinschaft Mathematik -- MAM-1}), held in Marburg in March 2017.

math.DG↗

arXiv2013

Twistor construction of asymptotically hyperbolic Einstein--Weyl spaces

Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ is a quotient of $Z$ by a holomorphic local $\mathbb{C}^*$ action. We prove that $T$ is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with $S$ as an asymptotic boundary.

math.DG↗