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Special Geometric Structures on Riemannian Manifolds Special Geometric Structures on Riemannian Manifolds Habilitationsschrift vorgelegt an der Fakult¨atf¨urMathematik der Universit¨atRegensburg von Mihaela Pilca 11. Januar 2016 2 Mentoren: Prof. Dr. Bernd Ammann, Universit¨atRegensburg Prof. Dr. Helga Baum, Humboldt-Universit¨atzu Berlin Prof. Dr. Simon Salamon, King's College London Aktuelle Adresse: Fakult¨atf¨urMathematik Universit¨atRegensburg D-93040 Regensburg Webseite: http://www.mathematik.uni-regensburg.de/pilca E-Mail: [email protected] 3 Contents Introduction 5 1 Homogeneous Clifford structures 19 1.1 Introduction . 19 1.2 Preliminaries . 21 1.3 The isotropy representation . 23 1.4 Homogeneous Clifford Structures . 32 2 Homogeneous almost quaternion-Hermitian manifolds 41 2.1 Introduction . 41 2.2 Preliminaries . 43 2.3 The classification . 44 A Root systems . 51 3 Eigenvalue Estimates of the spinc Dirac operator and harmonic forms on K¨ahler{Einsteinmanifolds 55 1 Introduction . 55 2 Preliminaries and Notation . 58 3 Eigenvalue Estimates of the spinc Dirac operator on K¨ahler{Einsteinman- ifolds . 63 4 Harmonic forms on limiting K¨ahler-Einsteinmanifolds . 70 4 The holonomy of locally conformally K¨ahlermetrics 77 1 Introduction . 77 2 Preliminaries on lcK manifolds . 80 3 Compact Einstein lcK manifolds . 82 4 The holonomy problem for compact lcK manifolds . 86 5 K¨ahlerstructures on lcK manifolds . 92 6 Conformal classes with non-homothetic K¨ahlermetrics . 95 5 Toric Vaisman Manifolds 107 1 Introduction . 107 2 Preliminaries . 108 3 Twisted Hamiltonian Actions on lcK manifolds . 112 4 Toric Vaisman manifolds . 115 5 Toric Compact Regular Vaisman manifolds . 122 6 Remarks on the product of harmonic forms 129 1 Introduction . 129 2 Geometric formality of warped product metrics . 130 3 Geometric formality of Vaisman metrics . 132 A Auxiliary results . 135 4 Introduction This habilitation thesis addresses several problems concerning special geometric struc- tures on Riemannian manifolds. The general framework into which this work fits is the question of finding \good" or \special" metrics on smooth manifolds, whereas these terms are considered in a large sense. The purpose of this introduction is, on the one hand, to briefly give an overview of the broader differential geometric context which made possible and motivated the study of the problems dealt with in this thesis, and on the other hand, to point out the relevance and the interrelation of the obtained results in this more general context. For a more precise presentation of the results we refer to the introduction of each chapter. There are different possible ways to define what \good" or \special" metrics should be. Very vaguely, one could require for such metrics to exist on \many" (compact) manifolds and at the same time to have distinguished properties which ensure that there are not \too many" of them on a fixed (compact) manifold. A very nice justification why Einstein metrics, defined as having constant Ricci curvature, can be considered as the \best" metrics on compact manifolds is given by A. Besse in the introduction of the book Einstein manifolds, [17]. Since there are just a few groups, namely those in the Berger-Simons list, which occur as non-generic irreducible holonomy groups of non-locally-symmetric metrics, the corresponding metrics can be also regarded as being \special". The reduction of the holonomy group can be characterized by the existence of certain parallel geometric structures compatible with the metric, like for instance a parallel complex structure in the K¨ahler case or a parallel spinor or form for the other holonomy groups. These conditions can be weakened in various ways in order to obtain more flexibility for the existence of such metrics. One possible generalization is to consider G-structures, defined as a reduction only of the structure group to some subgroup of the orthogonal group. For example, when G is the unitary group, we obtain an almost Hermitian structure. Another possibility is to look for the existence of particular spinors of a spin or more generally spinc structure, satisfying a weaker equation than the one for parallel spinors, like e.g. Killing spinors. In a different vein, distinguished metrics are also those having a lot of symmetries, such as toric K¨ahler metrics. When studying special geometric structures, an important question is to find topo- logical obstructions for a manifold to admit such structures. Apart from restrictions regarding for example the Betti numbers or the fundamental group, the issue of whether a manifold is formal or not has gained a lot of interest, in particular after the seminal 5 6 INTRODUCTION work [27] of P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, where they proved that compact K¨ahlermanifolds are formal. In the sequel we outline the topics of this work from the above mentioned point of view, namely as possible attempts to detect special metrics or special geometric struc- tures, like homogeneous Clifford structures, Einstein locally conformally K¨ahlermetrics, K¨ahler-Einsteinmetrics with special spinors, toric Vaisman metrics or formal metrics. Clifford structures. The notion of Clifford structure on a Riemannian manifold was introduced by A. Moroianu and U. Semmelmann, [59], and is in a certain sense dual to spin structures. Whereas in spin geometry, the spinor bundle is a representation space of the Clifford algebra bundle of the tangent bundle, in the case of (even) Clifford structures the roles are reversed, it is the tangent bundle of the manifold which becomes a representation space of the (even) algebra bundle of the so-called Clifford bundle (whose rank is called the rank of the Clifford structure). More precisely, a rank r Clifford structure is a rank r subbundle of the skew-symmetric endomorphisms of the tangent bundle, locally spanned by anti-commuting almost complex structures. As particular cases, we obtain for rank 1, almost Hermitian structures, and for rank 2, almost quaternion-Hermitian structures. Clifford structures are in particular G- structures, namely when there is a reduction of the structure group to Spin(r)·C(Pin(r)), where C(Pin(r)) denotes the centralizer in the special orthogonal group. The special class of so-called parallel Clifford structures, whose classification was carried out in [59], generalizes hyper-K¨ahlerstructures (r = 2) and quaternion-K¨ahlerstructures (r = 3). The next important subclass is represented by homogeneous Clifford structures. In [57] it is shown that the rank of an even homogeneous Clifford structure on a compact homogeneous manifold of non-vanishing Euler characteristic cannot exceed 16. More- over, for the ranks 9; 10; 12 and 16, it is proven that the only manifolds carrying such structures are the so-called Rosenfeld's elliptic projective planes, cf. [68]. The other extreme case, namely for rank 3, i. e. homogeneous almost quaternion-Hermitian man- ifolds of non-vanishing Euler characteristic, are classified in [58]. In fact, the only such 2 2 spaces are: the quaternionic symmetric spaces of Wolf (cf. [75]), S × S and the complex quadric SO(7)=U(3). The methods used in both articles [57] and [58] are of representation-theoretical nature, the main idea behind the classification being that the special configuration of the weights of the spinorial representation is not compatible with the usual integrality conditions of root systems. The remaining cases for intermediary ranks are still open. It would be particularly interesting if a non-symmetric space occurs for one of the left ranks. Some of the above examples of Clifford structures have been further investigated. F. M. Cabrera and A. Swann, [21], showed that on SO(7)=U(3), which is also the twistor space of the six sphere, there is exactly a one-dimensional family of SO(7)-invariant almost quaternion-Hermitian structures (with fixed volume). Moreover, they determined the types of their intrinsic torsion, according to the Gray-Hervella classes introduced in [36]. M. Parton and P. Piccinni [64] studied in more details the projective plane over the complex octonions, namely E6=(Spin(10) · U(1)), for which an explicit description of the Clifford bundle is given. Furthermore, using this description, they constructed 7 a canonical differential 8-form associated with the holonomy Spin(10) · U(1), which represents a generator of the cohomology ring. Spinc structures. Spin geometry played over the last fifty years a relevant role in both geometry and theoretical physics. Its importance was highlighted by the clas- sical Atiyah-Singer index formula for Dirac operators, which provided a fruitful link between the topology, geometry and analysis of the underlying manifold. Moreover, as mentioned above, most of the geometries with special holonomy also have a spinorial characterization. The condition for a manifold to be spin, which is equivalent to the vanishing of its second Stiefel-Whitney class, is restrictive for important classes of manifolds. For in- stance, the complex projective space is spin if and only if its complex dimension is odd. Therefore, one often considers a slight generalization as a complex analogue, namely the so-called spinc structures, when flexibility is gained by the existence of an auxiliary complex line bundle. The interest in these structures increased starting with the devel- opement of the Seiberg-Witten theory, [69]. Every orientable 4-dimensional manifold admits spinc-structures, as it was proven by P. Teichner and E. Vogt, [72], and previously by F. Hirzebruch and H. Hopf, [43], in the closed case. Also all almost complex man- ifolds carry canonical spinc structures. Special spinc spinors, generalizing for instance parallel or Killing spinors, play an important role in eigenvalue estimates of the spinc Dirac operator. M. Herzlich and A. Moroianu, [41], extended the classical estimate of T. Friedrich, [34], to compact Riemannian spinc manifolds. O. Hijazi, S. Montiel and F. Urbano, [42] constructed spinc structures on K¨ahler-Einsteinmanifolds of positive scalar curvature, which carry so-called K¨ahlerianKilling spinc spinors.
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