arXiv subjects
Publications and source records attributed to Marco Freibert.
We show that complex symplectic structures need not be preserved under small deformations, and we find sufficient conditions for this to happen. We study various cohomologies of compact complex symplectic manifolds, obtaining some topological obstructions to their existence.
In this article, we provide a general set-up for arbitrary linear Lie groups $H\leq \mathrm{GL}(n,\mathbb{R})$ which allows to characterise the almost Abelian Lie algebras admitting a torsion-free $H$-structure. In more concrete terms, using that an $n$-dimensional almost Abelian Lie algebra $\mathfrak{g}=\mathfrak{g}_f$ is fully determined by an endomorphism $f$ of $\mathbb{R}^{n-1}$, we give a description of the subspace $\mathcal{F}_{\mathfrak{h}}$ of all $f\in\mathrm{End}(\mathbb{R}^{n-1})$ for which $\mathfrak{g}_f$ admits a ``special'' torsion-free $H$-structure in terms of the image of a certain linear map. For large classes of linear Lie groups $H$, we are able to explicitly compute $\mathcal{F}_{\mathfrak{h}}$ and so give characterisations of the almost Abelian Lie algebras admitting a torsion-free $H$-structure. Our results reprove all the known characterisations of the almost Abelian Lie algebras admitting a torsion-free $H$-structure for different single linear Lie groups $H$ and extends them to big classes of linear Lie groups $H$. For example, we are able to provide characterisations in the case $n=2m$, $H\leq \mathrm{GL}(m,\mathbb{C})$ and $H$ either being a complex Lie group or being totally real, or in the case that $H$ preserves a pseudo-Riemannian metric. In many cases, we show that the space $\mathcal{F}_{\mathfrak{h}}$ coincides with what we call the \emph{characteristic subalgebra} $\tilde{\mathfrak{k}}_{\mathfrak{h}}$ associated to $\mathfrak{h}$, and that then the torsion-free condition is equivalent to the left-invariant flatness condition. In particular, we prove this to be the case if $H$ is a complex linear Lie group or if $\mathfrak{h}$ does not contain any elements of rank one or two and is either metric or totally real.
We continue the systematic study of left-invariant generalised Einstein metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra $\mathfrak{g}$, the unknowns of the system consist of a scalar product $g$ and a $3$-form $H$ on $\mathfrak{g}$ as well as a linear form $\delta$ on $\mathfrak{g}\oplus\mathfrak{g}^*$. As in arXiv:2206.01157, the Lie bracket of $\mathfrak{g}$ is considered part of the unknowns. In the Riemannian case, we show that the generalised Einstein condition always reduces to the commutator ideal and we provide a full classification of solvable generalised Einstein Lie groups. In the Lorentzian case, under the additional assumption $\delta=0$, we classify -- up to one case -- all almost Abelian generalised Einstein Lie groups. We then particularize to four dimensions and provide a full classification of generalised Einstein Riemannian Lie groups as well as generalised Einstein Lorentzian Lie groups with $\delta =0$ and non-degenerate commutator ideal.
We present two constructions of complex symplectic structures on Lie algebras with large abelian ideals. In particular, we completely classify complex symplectic structures on almost abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)k\"ahler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.
It has been conjectured by Fino and Vezzoni that a compact complex manifold admitting both a compatible SKT and a compatible balanced metric also admits a compatible K\"ahler metric. Using the shear construction and classification results for two-step solvable SKT Lie algebras from our previous work, we prove this conjecture for compact two-step solvmanifolds endowed with an invariant complex structure which is either (a) of pure type or (b) of dimension six. In contrast, we provide two counterexamples for a natural generalisation of this conjecture in the homogeneous invariant setting. As part of the work, we obtain further classification results for invariant SKT, balanced and K\"ahler structures on two-step solvable Lie groups. In particular, we give the full classification of left-invariant SKT structures on two-step solvable Lie groups in dimension six.
A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit a left-invariant closed $\mathrm{G}_2$-eigenform for the Laplacian and that any compact solvmanifold $Γ\backslash G$ arising from $G$ does not admit an (invariant) exact $\mathrm{G}_2$-structure. We also classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact $\mathrm{G}_2$-structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras $\mathfrak{h}$ admitting an exact $\mathrm{SL}(3,\mathbb{C})$-structure $ρ$ or a half-flat $\mathrm{SU}(3)$-structure $(ω,ρ)$ with exact $ρ$, respectively.
We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for $\mathfrak{g}$ almost Abelian, for derived algebra $\mathfrak{g}'$ of codimension 2 and not $J$-invariant, for $\mathfrak{g}'$ totally real, and for $\mathfrak{g}'$ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.
We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.
We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplectic oxidation to classify eight-dimensional nilpotent complex symplectic Lie algebras.
We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension 6 is treated.
The twist construction is a method to build new interesting examples of geometric structures with torus symmetry from well-known ones. In fact it can be used to construct arbitrary nilmanifolds from tori. In our previous paper, we presented a generalization of the twist, a shear construction of rank one, which allowed us to build certain solvable Lie algebras from $\mathbb{R}^n$ via several shears. Here, we define the higher rank version of this shear construction using vector bundles with flat connections instead of group actions. We show that this produces any solvable Lie algebra from $\mathbb{R}^n$ by a succession of shears. We give examples of the shear and discuss in detail how one can obtain certain geometric structures (calibrated $\mathrm{G}_2$, co-calibrated $\mathrm{G}_2$ and almost semi-Kähler) on two-step solvable Lie algebras by shearing almost Abelian Lie algebras. This discussion yields a classification of calibrated $\mathrm{G}_2$-structures on Lie algebras of the form $(\mathfrak{h}_3\oplus \mathbb{R}^3)\rtimes \mathbb{R}$.
The Hitchin flow constructs eight-dimensional Riemannian manifolds (M,g) with holonomy in Spin(7) starting with a cocalibrated G_2-structure on a seven-dimensional manifold. As Sp(2)\subseteq SU(4)\subseteq Spin(7), one may also obtain Calabi-Yau fourfolds or hyperKähler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics with Hol(g)\subseteq SU(4) but Hol(g)\neq Sp(2). We investigate when we actually get Hol(g)=SU(4) and obtain so many new explicit examples of Calabi-Yau fourfolds. The results rely on the connection between cocalibrated G_2-structures and hypo SU(3)-structures and between the Hitchin and the hypo flow and on a systematic study of hypo SU(3)-structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo SU(3)-structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or Sp(2)-holonomy or do get Hol(g)=SU(4) for the Riemannian manifolds obtained by the hypo flow with initial values in some other intrinsic torsion classes.
The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geometric structures that may be obtained from the shear construction.
We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$, respectively, which are in general geodesically incomplete. Generalizing results of Conti, we prove that for large classes of solvable Lie groups $G$ these manifolds cannot be completed: a complete Riemannian manifold with parallel $SU(3)$-, $G_2$- or $Spin(7)$-structure which is of cohomogeneity one with respect to $G$ is flat, and has no singular orbits. We furthermore classify, on the non-compact Lie group $SL(2,C)$, all half-flat $SU(3)$-structures which are bi-invariant with respect to the maximal compact subgroup $SU(2)$ and solve the Hitchin flow for these initial values. It turns out that often the flow collapses to a smooth manifold in one direction. In this way we recover an incomplete cohomogeneity-one Riemannian metric with holonomy equal to $G_2$ on the twisted product $SL(2,C)\times_{SU(2)} C^2$ described by Bryant and Salamon.
In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the first example of a Ricci-flat calibrated G_2^*-structure on a compact manifold whose holonomy is not contained in G_2^*. Moreover, we get examples of non-flat parallel G_2^*-structures on almost Abelian Lie algebras g. We give a full classification of these G_2^*-structures if g is additionally nilpotent.
Cocalibrated G_2-structures are structures naturally induced on hypersurfaces in Spin(7)-manifolds. Conversely, one may start with a seven-dimensional manifold M endowed with a cocalibrated G_2-structure and construct via the Hitchin flow a Spin(7)-manifold which contains M as a hypersurface. In this article, we consider left-invariant cocalibrated G_2-structures on Lie groups G which are a direct product G=G_4\times G_3 of a four-dimensional Lie group G_4 and a three-dimensional Lie group G_3. We achieve a full classification of the Lie groups G=G_4\times G_3 which admit a left-invariant cocalibrated G_2-structure.
Half-flat SU(3)-structures are the natural initial values for Hitchin's evolution equations whose solutions define parallel G_2-structures. Together with the results of arXiv:0912.3486v1, the results of this article completely solve the existence problem of left-invariant half-flat SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras which refines and clarifies the existing classification of five-dimensional Lie algebras.
This article can be viewed as a continuation of the articles arXiv:0912.3486 and arXiv:1012.3714 where the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradical which admit a half-flat SU(3)-structure. As an important step of the proof, a considerable refinement of the classification of six-dimensional Lie algebras with five-dimensional non-Abelian nilradical is established. Additionally, it is proved that all non-solvable six-dimensional Lie algebras admit half-flat SU(3)-structures.