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Anna Fino

Publications and source records attributed to Anna Fino.

At least 19 recordsLinked to original sources
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↗
arXiv2026

A note on the pluriclosed flow on balanced manifolds with $c_1=0$

We conjecture that on any compact balanced manifold $(M, \omega_B)$ with $c_{1}(M)=0$, the pluriclosed flow admits long-time solutions $\omega_{t}$ for every initial pluriclosed metric, and that $\omega_{t}$ converges smoothly to a K\"ahler metric as $t \to \infty$. We verify that this phenomenon occurs when $M$ is a compact quotient of a Lie group by a discrete subgroup, the background metric $\omega_{B}$ is invariant with vanishing Chern--Ricci form, and the initial metric $\omega_{0}$ is invariant. In particular, this provides new evidences for the Fino-Vezzoni conjecture.

math.DG↗
arXiv2026

$p$-K\"ahler structures on fibrations and reductive Lie groups

We investigate the existence of $p$-K\"ahler structures on two classes of complex manifolds: on quasi-regular fibrations, with particular emphasis on complex homogeneous spaces, and on reductive Lie groups endowed with invariant complex structures. In the latter setting, we construct non-regular complex structures on the Lie algebras $\mathfrak{sl}(2m-1,\mathbb{R})$ for $m \ge 2$ and show that these structures admit compatible balanced metrics, providing new explicit examples of balanced manifolds.

math.DG↗
arXiv2026

New Solutions to the $G_2$ Hull-Strominger System via torus fibrations over $K3$ orbifolds

Using torus fibrations over K3 orbisurfaces, we construct new smooth solutions to the $G_2$ Hull-Strominger system. These manifolds arise as total spaces of principal $T^3$ (orbi)bundles over singular K3 surfaces. Our construction is based on the choice of three divisors on a singular K3 surface that are primitive with respect to a particular K\"ahlermetric. The stable bundle is obtained via an adaptation of the Serre construction to the singular setting.

math.DG↗
arXiv2026

On a Modification of the Twistor Space

In the paper we construct a modification $S(M)$ of the twistor space of a K\"ahler scalar flat surface $M$ and study its complex-geometric and metric properties. In particular, we construct complete balanced metrics on $S(M)$ and show that $S(M)$ can not be K\"ahler when $M$ is a compact simple hyperk\"ahler manifold.

math.DG↗
arXiv2025

$\del\delbar$-Lemma and Bott-Chern cohomology of twistor spaces

In the paper we study the Bott-Chern and Aeppli cohomologies of the twistor space of a compact self-dual 4-manifold and we characterize the validity of the $\partial \overline \partial$-lemma. We also compute explicitly the Dolbeault cohomology of the twistor space $Z$ of the flat $4$-dimensional torus, which is known to not satisfy the $\partial\overline{\partial}$ lemma.

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↗
arXiv2025

Some remarks on strong $\mathrm{G}_2$-structures with torsion

A $\mathrm{G}_2$-structure on a $7$-manifold $M$ is called a $\mathrm{G}_2T$-structure if $M$ admits a $\mathrm{G}_2$-connection $\nabla^T$ with totally skew-symmetric torsion $T_\varphi$. If furthermore, $T_\varphi$ is closed then it is called a strong $\mathrm{G}_2T$-structure. In this paper we investigate the geometry of (strong) $\mathrm{G}_2T$-manifolds in relation to its curvature, $S^1$ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of $\nabla^T$ and give an equivalent characterisation in terms of geometric properties of the $\mathrm{G}_2$ Lee form. Analogous results are also obtained for almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the $S^1$ reduction by the dual of the $\mathrm{G}_2$ Lee form, we show that Ricci-flat strong $\mathrm{G}_2T$-structures correspond to solutions of the $\mathrm{SU}(3)$ heterotic system on certain almost Hermitian half-flat $6$-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong $\mathrm{G}_2T$-structures with $\nabla^T$ not Ricci flat. Lastly, we classify $\mathrm{G}_2$-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.

math.DG↗
arXiv2025

Fibrations Over Singular K3 Surfaces and New Solutions to the Hull-Strominger System

Using fibrations over K3 orbisurfaces we construct new smooth solutions to the Hull-Strominger system. In particular, we prove that, for $4 \leq k \leq 22$ and $5 \leq r\leq 22$, the smooth manifolds $S^1\times \sharp_k(S^2\times S^3)$ and $\sharp_r (S^2 \times S^4) \sharp_{r+1} (S^3 \times S^3)$, have a complex structure with trivial canonical bundle and admit a solution to the Hull-Strominger system.

math.DG↗
arXiv2024

On the type of generalized hypercomplex structures

The generalized hypercomplex structures defined within the framework of generalized geometry include hypercomplex and holomorphic symplectic structures as particular cases. They have a $S^2$-family of generalized complex structures, and in this paper we study the types of these structures and the corresponding twistor space. We show that there are generalized hypercomplex structures on the $4n$-dimensional tori, which do not contain a structure of maximal (complex) type. Moreover, we show that the Kodaira-Thurston surface which has a holomorphic symplectic structure, admits also a generalized hypercomplex structure in which all generalized complex structures are of type $1$.

math.DG↗
arXiv2024

On the nu-invariant of two-step nilmanifolds with closed G2-structure

For every non-vanishing spinor field on a Riemannian spin seven-manifold, Crowley, Goette, and Nordstr\"om defined the so-called $\nu$-invariant. This is an integer modulo $48$ that detects connected components of the moduli space of $\mathrm G_2$-structures on any seven-dimensional oriented spin manifold. The $\nu$-invariant can be defined in terms of Mathai--Quillen currents, harmonic spinors, and $\eta$-invariants of spin Dirac and odd-signature operator. We compute these data for certain families of left-invariant closed $\mathrm G_2$-structures on compact two-step nilmanifolds with their natural spin structure. Specifically, we establish the existence of non-invariant harmonic spinors and determine the parity of the dimension of the space of harmonic spinors. We deduce the vanishing of $\nu$ on invariant harmonic spinors.

math.DG↗
arXiv2024

On the existence of balanced metrics of Hodge-Riemann type

In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-K\"ahler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-K\"ahler. Then, we focus on the case of compact quotients of Lie groups by lattices, endowed with an invariant complex structure. In particular, we prove non existence results on non-K\"ahler complex parallelizable manifolds and some classes of solvmanifolds, and we show that the only nilmanifolds admitting invariant structures of this type are tori. Finally, we construct the first non-K\"ahler example of a Hodge-Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by $\mathbb C$.

math.DG↗
arXiv2024

SKT solvable Lie algebras with codimension two nilradical

In the present paper we study SKT and generalized K\"ahler 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 K\"ahler 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

A note on $p$-Kähler structures on compact quotients of Lie groups

A $p$-Kähler structure on a complex manifold of complex dimension $n$ is given by a $d$-closed transverse real $(p,p)$-form. In the paper we study the existence of $p$-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of $p$-Kähler structures on nilmanifolds, with a focus on the case $p =2$ and complex dimension $n = 4$. Moreover, we prove that a $(n-2)$-Kähler almost abelian solvmanifold of complex dimension $n\geq3$ has to be Kähler.

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↗