We prove that, given a certain isometric action of a two-dimensional Abelian group A on a quaternionic Kähler manifold M which preserves a submanifold N\subset M, the quotient M'=N/A has a natural Kähler structure. We verify that the assumptions on the group action and on the submanifold N\subset M are satisfied for a large class of examples obtained from the supergravity c-map. In particular, we find that all quaternionic Kähler manifolds M in the image of the c-map admit an integrable complex structure compatible with the quaternionic structure, such that N\subset M is a complex submanifold. Finally, we discuss how the existence of the Kähler structure on M' is required by the consistency of spontaneous {\cal N}=2 to {\cal N}=1 supersymmetry breaking.
We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an {é}tale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of $k$-dimensional quaternionic projective space by a $(k-1)$-torus.
In this paper, metric reduction in generalized geometry is investigated. We show how the Bismut connections on the quotient manifold are obtained from those on the original manifold. The result facilitates the analysis of generalized K$\ddot{a}$hler reduction, which motivates the concept of metric generalized principal bundles and our approach to construct a family of generalized holomorphic line bundles over $\mathbb{C}P^2$ equipped with some non-trivial generalized K$\ddot{a}$hler structures.
We classify those manifolds mentioned in the title which have finite topological type. Namely we show any such connected M is isomorphic to a hyperkaehler quotient of a flat quaternionic vector space by an abelian group. We also show that a compact connected and simply connected 3-Sasakian manifold of dimension 4n-1 whose isometry group has rank n+1 is isometric to a 3-Sasakian quotient of a sphere by a torus. As a corollary, a compact connected quaternion-Kaehler 4n-manifold with positive scalar curvature and isometry group of rank n+1 is isometric to the quaternionic projective space or the …
The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galicki and Piccinni in \emph{$3-$Sasakian geometry, nilpotents orbits, and exeptional quotients}. We construct explicitly our new examples as quaternion-K$\ddot{\mathrm{a}}$hler reductions of the quaternion K$\ddot{\mathrm{a}}$hler Grassmannian $Gr_4(\mathbb{R}^8)$ by an isometric act…
We consider the moduli space M_r of polygons with fixed side lengths in five-dimensional eucledian space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between M_r and a weighted quotient of the n-fold product of the quaternionic projective line HP^1 by the diagonal PSL(2,H)-action. We explore the relation between M_r and the fixed point set of an anti-symplectic involution on a GIT quotient Gr(2,4)^n/SL(4,C). We generalize the Gel'fand-MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our …
Let $M^{4n}$ be a complete quaternionic Kähler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $λ_1(M)$ of the Laplacian which is $λ_1(M)\le (2n+1)^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\mathbb{R}\times N$ with N given by a compact manifold. Moreover, if $M$ is of bounded curvature, $M$ is covered by the quaterionic hyperbolic space $\mathbb{QH}^n$ and $N$ is a compact quotient of the generalized Heisenberg group. When $λ_1(M)\ge \frac{8(n+2)}3$, we also prove that $M$ must have onl…
Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient o…
Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by studying a natural non-associative algebra $\mathfrak{m}$ comming from the affine structure of $Sp(1) \backslash H$. We compute the automorphism group of $\mathfrak{m}$ and as a consecuence of this, we are able to compute the isometry group of $Sp(1) \backslash H$ at least up to connected components. Using these results, we obtain a uniqueness result on the structu…
We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonnegative Ricci and almost nonnegative sectional curvature. Moreover, if N has finite fundamental group, then N admits also metrics of positive Ricci curvature. Particular examples include infinite families of simply connected manifolds with the rational cohomology rings and integr…
We show that $3$-Sasaki structures admit a natural description in terms of projective differential geometry. This description provides a concrete link between $3$-Sasaki structures and several other geometries and constructions via a single unifying picture. First we establish that a $3$-Sasaki structure may be understood as a projective structure equipped with a certain holonomy reduction to the (possibly indefinite) unitary quaternionic group $\textrm{Sp}(p,q)$, namely a parallel hyperkähler structure on the projective tractor bundle satisfying a particular genericity condition. For the conv…
Let $G$ be a connected Lie group and $Γ\subset G$ a lattice. Connection curves of the homogeneous space $M=G/Γ$ are the orbits of one parameter subgroups of $G$. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a \textit{finite} set $B \subset M\setminus \{m_1, m_2 \}$ such that every connecting curve joining $m_1$ and $m_2$ intersects $B$. The homogeneous space $M$ is \textit{blockable} if every pair of points in $M$ can be blocked. \par In this paper we investigate blocking properties of $M_n=SL(n,R)/Γ$, where $Γ=SL(n,Z)$ is the integer lattice. We focus on $M_2$ and show that t…
Given a Kähler manifold $M$ endowed with a Hamiltonian Killing vector field $Z$, we construct a conical Kähler manifold $\hat{M}$ such that $M$ is recovered as a Kähler quotient of $\hat{M}$. Similarly, given a hyper-Kähler manifold $(M,g,J_1,J_2,J_3)$ endowed with a Killing vector field $Z$, Hamiltonian with respect to the Kähler form of $J_1$ and satisfying $\mathcal{L}_ZJ_2= -2J_3$, we construct a hyper-Kähler cone $\hat{M}$ such that $M$ is a certain hyper-Kähler quotient of $\hat{M}$. In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergr…
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and…
We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^1$. We partition the class of such nearly-Ka}hler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete. We also derive a quasilinear elliptic PDE system on the 2-dimen…
We describe an action of $U(1)\times Sp(1)$ on $HP^7$ that allows to obtain a $Z_2$-quotient of CAYLEY (the Grassmannian of oriented 4-planes in $R^8$ closed under the three-fold cross product) by quaternionic Kähler reduction.
We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained as quaternionic quotients of the Wolf spaces ${\mathbb H}P^2$, ${\mathbb H}H^2$, $SU(4)/S(U(2)U(2))$, and $SU(2,2)/S(U(2)U(2))$ are always Hermitian.
A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the fundamental domain of any lattice contains a ball whose radius depends only on the group itself. A direct consequence is a positive minimum volume for orbifolds modeled on the corresponding symmetric space. However, sharp bounds are known only for hyperbolic orbifolds of dimensions two and three, and recently for quaternionic hyperbolic orbifolds of all dimensions. A…
The tree-level q-map assigns to a projective special real (PSR) manifold of dimension $n-1\geq 0$, a quaternionic Kähler (QK) manifold of dimension $4n+4$. It is known that the resulting QK manifold admits a $(3n+5)$-dimensional universal group of isometries (i.e. independently of the choice of PSR manifold). On the other hand, in the context of Calabi-Yau compactifications of type IIB string theory, the classical hypermultiplet moduli space metric is an instance of a tree-level q-map space, and it is known from the physics literature that such a metric has an $\mathrm{SL}(2,\mathbb{R})$ group…
We study riemannian coverings $φ: \widetilde{M} \to Γ\backslash \widetilde{M}$ where $\widetilde{M}$ is a normal homogeneous space $G/K_1$ fibered over another normal homogeneous space $M = G/K$ and $K$ is locally isomorphic to a nontrivial product $K_1\times K_2$. The most familiar such fibrations $π: \widetilde{M} \to M$ are the natural fibrations of Stieffel manifolds $SO(n_1 + n_2)/SO(n_1)$ over Grassmann manifolds $SO(n_1 + n_2)/[SO(n_1)\times SO(n_2)]$ and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar…
