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Alejandro Gil-García

Publications and source records attributed to Alejandro Gil-García.

15 recordsLinked to original sources
arXiv2026

Kenmotsu manifolds and spin-c Killing spinors

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with an imaginary Killing function $iμ$ if and only if it is an exact $μ$-Kenmotsu manifold, thereby obtaining an extension of a recent result by the first named author that, under the purity assumption, does not require simple connectivity or completeness. We then reinterpret $μ$-Kenmotsu manifolds in terms of metric connections with vectorial torsion and give a second proof of this characterization, combining the theory of complex spinorial forms with the theory of metric connections with torsion. Finally, we describe the global structure of exact $μ$-Kenmotsu manifolds by means of Morse-Bott theory.

math.DG↗
arXiv2026

Torsion parallel pure spinors on neutral manifolds

We study irreducible real pure spinors on pseudo-Riemannian manifolds of neutral signature using the theory of real spinorial forms. We prove that the square of such a spinor is a decomposable differential form of middle degree satisfying a natural duality condition. In signature $(4,4)$, we show that non-pure spinors correspond to $\mathrm{Spin}_0(4,3)$-structures, yielding an intrinsic algebraic characterization of these structures. In addition, we characterize real pure spinors parallel with respect to metric connections with torsion in terms of an equivalent differential system for their squares. As an application, we study left-invariant supersymmetric solutions of the NS-NS supergravity system on certain four-dimensional Lie groups.

math.DG↗
arXiv2026

Sasakian manifolds and spin-c Killing spinors

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with a real Killing constant $α\in\mathbb{R}^{\ast}$ if and only if it is $α$-Sasakian, thereby obtaining an extension of a well-known result by A. Moroianu that, under the purity assumption, does not require simple connectivity or completeness.

math.DG↗
arXiv2026

Almost abelian pseudo-Kähler Lie algebras

We study invariant pseudo-Kähler structures on a solvmanifold $G$ such that the Lie algebra $\mathfrak{g}$ is almost abelian, that is $\mathfrak{g}=\mathfrak{h}\rtimes\mathbb{R}$, with $\mathfrak{h}$ abelian; comparing with the positive-definite case, an additional situation occurs, corresponding to the ideal $\mathfrak{h}$ being degenerate. We obtain a classification up to unitary isomorphism in all dimensions. We deduce that every nilpotent almost abelian Lie algebra endowed with a complex structure also admits a compatible pseudo-Kähler structure, and prove that this is no longer true for general almost abelian Lie algebras; indeed, we classify all the almost abelian Lie algebras that admit a complex structure and a symplectic structure but no compatible pseudo-Kähler metric. We study the curvature of the metrics we have obtained, and use some of them to construct Einstein pseudo-Kähler metrics in two dimensions higher.

math.DG↗
arXiv2026

Complex spinorial forms, Brinkmann four-manifolds, and self-dual bundle gerbes

We develop the differential theory of complex spinorial forms associated with irreducible complex spinors across all dimensions and signatures. This framework enables the study of constrained parallelicity conditions for irreducible complex spinors by reformulating them as equivalent differential systems for exterior forms within a prescribed semi-algebraic body of the Kähler-Atiyah bundle. To illustrate this approach, we first apply it to the spin-c Killing spinor equation in low dimensions, refining existing results by relaxing standard assumptions of simply connectedness and completeness. Then, we proceed to apply our framework to supersymmetry conditions in supergravity, and we prove that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts. Finally, we study the quasi-supersymmetric solutions of six-dimensional minimal supergravity, defined by a system that couples a self-dual curving on a bundle gerbe to a Lorentzian metric with an irreducible chiral spinor parallel under a metric connection with totally skew-symmetric torsion given by the curvature of the aforementioned curving. Along the way, we prove that a Lorentzian six-manifold admits a skew-torsion parallel spinor with an integrable screen bundle only if it admits a foliation whose leaves are locally conformally Kähler complex surfaces.

math.DG↗
arXiv2026

Left-invariant harmonic spinors on three-dimensional Lie groups

We study the existence of left-invariant harmonic spinors on three-dimensional Lie groups equipped with a left-invariant pseudo-Riemannian metric. An existing formula for the spin Dirac operator acting on left-invariant spinors in the Riemannian setting is revised and specialised to our cases, in particular to almost Abelian Lie algebras. Focussing on dimension two and three, we find equivalent conditions for the Lie groups to admit left-invariant harmonic spinors in terms of constraints on the structure equations of the corresponding Lie algebras. We then identify those metrics (up to automorphism) carrying left-invariant harmonic spinors in each case.

math.DG↗
arXiv2025

The algebraic square of an irreducible complex spinor

We characterize, in every dimension and signature, the algebraic squares of an irreducible complex spinor as a pair of exterior forms satisfying a prescribed system of algebraic relations that we present in terms of the geometric product of the underlying quadratic vector space. As a result, we obtain a general correspondence between irreducible complex spinors and algebraically constrained exterior forms, which clarifies the subtle relationship between spinors and exterior forms and contributes towards the understanding of spinors as the square root of geometry. We use this formalism to construct the squares of an irreducible complex spinor in Euclidean dimensions up to six, and also to construct the squares of a generic, possibly non-pure and non-unit, irreducible complex chiral spinor in eight Euclidean dimensions. Elaborating on this result, we consider a natural notion of spinorial instanton that we study for connections on a principal bundle with a complex structure group as well as for curvings of a $\mathbb{C}^{\ast}$-bundle gerbe defined on a Lorentzian six-manifold.

math.DG↗
arXiv2025

Born Lie algebras

We show that every Born Lie algebra can be obtained by the bicross product construction starting from two pseudo-Riemannian Lie algebras. We then obtain a classification of all Lie algebras up to dimension four and all six-dimensional nilpotent Lie algebras admitting an integrable Born structure. Finally, we study the curvature properties of the pseudo-Riemannian metrics of the integrable Born structures obtained in our classification results.

math.DG↗
arXiv2025

Parallel spinors for $\mathrm{G}_2^*$ and isotropic structures

We obtain a correspondence between irreducible real parallel spinors on pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions of an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the Kähler-Atiyah bundle of $(M,g)$. Applying this general framework, we obtain an intrinsic algebraic characterization of $\mathrm{G}_2^*$-structures as well as the first explicit description of isotropic irreducible spinors in signature $(4,3)$ that are parallel under a general connection on the spinor bundle. This description is given in terms of a coherent system of mutually orthogonal and isotropic one forms and follows from the characterization of the stabilizer of an isotropic spinor as the stabilizer of a highly degenerate three-form that we construct explicitly. Using this result, we show that isotropic spinors parallel under a metric connection with torsion exist when the connection preserves the aforementioned coherent system. This allows us to construct a natural class of metrics of signature $(4,3)$ on $\mathbb{R}^7$ that admit spinors parallel under a metric connection with torsion.

math.DG↗
arXiv2025

Quaternionic Kähler manifolds fibered by solvsolitons

This paper is concerned with the geometry of principal orbits in quaternionic Kähler manifolds $M$ of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic Kähler symmetric spaces associated with the simple Lie groups of type A by the one-loop deformation. We prove that for zero deformation parameter the principal orbits form a fibration by solvsolitons (nilsolitons if $4n=\dim M=4$). The underlying solvable group is non-unimodular if $n>1$ and is the Heisenberg group if $n=1$. We show that under the deformation, the hypersurfaces remain solvmanifolds but cease to be Ricci solitons.

math.DG↗
arXiv2024

Pseudo-Kähler and hypersymplectic structures on semidirect products

We study left-invariant pseudo-Kähler and hypersymplectic structures on semidirect products $G\rtimes H$; we work at the level of the Lie algebra $\mathfrak{g}\rtimes\mathfrak{h}$. In particular we consider the structures induced on $\mathfrak{g}\rtimes\mathfrak{h}$ by existing pseudo-Kähler structures on $\mathfrak{g}$ and $\mathfrak{h}$; we classify all semidirect products of this type with $\mathfrak{g}$ of dimension $4$ and $\mathfrak{h}=\mathbb{R}^2$. In the hypersymplectic setting, we consider a more general construction on semidirect products. We construct a large class of hypersymplectic Lie algebras whose underlying complex structure is not abelian as well as non-flat hypersymplectic metrics on $k$-step nilpotent Lie algebras for every $k\geq3$.

math.DG↗
arXiv2024

Moduli spaces of (co)closed $\mathrm{G}_2$-structures on nilmanifolds

We compute the dimensions of some moduli spaces of left-invariant closed and coclosed $\mathrm{G}_2$-structures on 7-dimensional nilmanifolds, showing that they are not related to the third Betti number. We also prove that, in contrast to the case of closed $\mathrm{G}_2$-structures, the group of automorphisms of a coclosed $\mathrm{G}_2$-structure is not necessarily abelian.

math.DG↗
arXiv2024

Symmetries of one-loop deformed q-map spaces

Q-map spaces form an important class of quaternionic Kähler manifolds of negative scalar curvature. Their one-loop deformations are always inhomogeneous and have been used to construct cohomogeneity one quaternionic Kähler manifolds as deformations of homogeneous spaces. Here we study the group of isometries in the deformed case. Our main result is the statement that it always contains a semidirect product of a group of affine transformations of $\mathbb{R}^{n-1}$ with a Heisenberg group of dimension $2n+1$ for a q-map space of dimension $4n$. The affine group and its action on the normal Heisenberg factor in the semidirect product depend on the cubic affine hypersurface which encodes the q-map space.

math.DG↗
arXiv2021

Symmetric and skew-symmetric complex structures

On a complex manifold $(M,J)$, we interpret complex symplectic and pseudo-Kähler structures as symplectic forms with respect to which $J$ is, respectively, symmetric and skew-symmetric. We classify complex symplectic structures on 4-dimensional Lie algebras. We develop a method for constructing hypersymplectic structures from the above data. This allows us to obtain an example of a hypersymplectic structure on a 4-step nilmanifold.

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