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Anna Fino

Publications and source records attributed to Anna Fino.

At least 19 recordsLinked to original sources
arXiv2026

Strong $G_2$-structures with torsion: a survey and new examples

This article has two main parts. The first consists of a brief survey of $G_2$-structures with torsion, namely $G_2$-structures admitting a $G_2$-connection with totally skew-symmetric torsion. We focus primarily on the strong condition that the associated torsion $3$-form is closed. We review their basic properties and their connections with the heterotic $G_2$ system, generalized geometry, and six-dimensional geometry. In the second part, we construct two complete cohomogeneity-one strong $G_2T$-structures on $\mathbb{R}\times S^3\times S^3$ with non-zero closed and co-closed characteristic torsion, and non-vanishing characteristic Ricci tensor. Both have an asymptotically cylindrical end; at the other end, the first is asymptotically conical (AC), with a torsion-free $G_2$ cone as its asymptotic model, while the second is asymptotically locally conical (ALC), with a constant-length circle bundle over a Calabi--Yau cone as its asymptotic model. We investigate the induced $SU(3)$-structures on their circle quotients and show that they are globally conformally almost Kähler and globally conformally Kähler, respectively. To the best of our knowledge, these provide the first complete strong $G_2T$-structures whose underlying metrics are not locally Riemannian products.

math.DG↗
arXiv2026

Almost Hermitian geometry and generalized Ricci flow

We study almost Hermitian manifolds whose lowered Nijenhuis tensor is totally skew-symmetric and whose characteristic (Bismut) torsion is closed, the almost-SKT condition. This extends the SKT (pluriclosed) condition beyond the integrable setting. For a fixed almost complex structure, these constraints are linear in the metric. We introduce an overdetermined ellipticity condition which, on compact manifolds, makes the admissible infinitesimal deformation space finite-dimensional and yields short-time existence for a constrained Bismut--Ricci flow. In real dimension six, almost-SKT structures satisfy a dichotomy: either the almost complex structure is integrable, or the skew Nijenhuis tensor is nowhere vanishing with constant norm and the Bismut Ricci form vanishes. Hence, on compact non-integrable almost-SKT six-manifolds, the constrained Bismut--Ricci flow is stationary. In dimension eight, ellipticity always fails, while in higher dimensions Nijenhuis rigidity holds on an open dense locus. In arbitrary dimension, we characterize when total skew-symmetry of the lowered Nijenhuis tensor determines the compatible metric pointwise up to scale. On compact manifolds, with closed torsion, this implies uniqueness of the almost-SKT metric up to scaling; the constrained Bismut--Ricci flow is then an explicit homothetic solution. We also relate this flow to generalized Ricci flow and characterize the infinitesimal obstruction to lifting the evolution when the almost complex structure varies. Finally, we construct non-integrable almost-SKT structures with nowhere vanishing Nijenhuis tensor and non-flat characteristic connection on compact six-manifolds, including $S^1\times S^2\times S^3$ and $S^3\times S^3$. In these families, the metric is fixed and the compatible almost complex structures form an $S^2$-family. These structures have vanishing Bismut Ricci form and are therefore almost-CYT.

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 Kählermetric. The stable bundle is obtained via an adaptation of the Serre construction to the singular setting.

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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-Kähler-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 $S^1$.

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arXiv2026

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

We conjecture that on any compact balanced manifold $(M, ω_B)$ with $c_{1}(M)=0$, the pluriclosed flow admits long-time solutions $ω_{t}$ for every initial pluriclosed metric, and that $ω_{t}$ converges smoothly to a Kähler 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 $ω_{B}$ is invariant with vanishing Chern--Ricci form, and the initial metric $ω_{0}$ is invariant. In particular, this provides new evidences for the Fino-Vezzoni conjecture.

math.DG↗
arXiv2026

$\del\delbar$-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↗
arXiv2026

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 Kähler 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↗
arXiv2026

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 Nordström defined the so-called $ν$-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 $ν$-invariant can be defined in terms of Mathai--Quillen currents, harmonic spinors, and $η$-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 $ν$ on invariant harmonic spinors.

math.DG↗
arXiv2026

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-Kähler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-Kähler. 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-Kähler 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-Kähler 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↗
arXiv2026

$p$-Kähler structures on fibrations and reductive Lie groups

We investigate the existence of $p$-Kähler 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

On a Modification of the Twistor Space

In the paper we construct a modification $S(M)$ of the twistor space of a Kähler 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 Kähler when $M$ is a compact simple hyperkähler manifold.

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↗
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 $T_φ$. If furthermore, $T_φ$ 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

Optical geometries

We study the notion of optical geometry, defined to be a Lorentzian manifold equipped with a null line distribution, from the perspective of intrinsic torsion. This is an instance of a non-integrable version of holonomy reduction in Lorentzian geometry. These generate congruences of null curves, which play an important rôle in general relativity. Conformal properties of these are investigated. We also extend this concept to generalised optical geometries as introduced by Robinson and Trautman.

math.DG↗
arXiv2025

Special Hermitian structures on suspensions

Motivated by the construction based on topological suspension of a family of compact non-Kähler complex manifolds with trivial canonical bundle given by L. Qin and B. Wang in [QW], we study toric suspensions of balanced manifolds by holomorphic automorphisms. In particular, we show that toric suspensions of Calabi-Yau manifolds are balanced. We also prove that suspensions associated with hyperbolic automorphisms of hyperkähler manifolds do not admit any pluriclosed, astheno-Kähler or p-pluriclosed Hermitian metric. Moreover, we consider natural extensions for hypercomplex manifolds, providing some explicit examples of compact holomorphic symplectic and hypercomplex non-Kähler manifolds. We also show that a modified suspension construction provides examples with pluriclosed metrics.

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

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.

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