Scalar-flat K?hler orbifolds via quaternionic-complex reduction

We prove that any asymptotically locally Euclidean scalar-flat K?hler 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.     

Toric Anti-self-dual 4-manifolds Via Complex Geometry

Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the ?ech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kähler representatives are described; each such class contains a circle of Kähler metrics. This gives new local examples of scalar flat Kähler surfaces and generalises work of Joyce [Duke. Math. J. 77(3), 519–552 (1995)] who considered the case where the distribution orthogonal to the torus action is integrable.     

Non-minimal scalar-flat Kähler surfaces and parabolic stability

A new construction is presented of scalar-flat Kähler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that \(\mathbb{CP}^2\) blown up at 10 suitably chosen points, admits a scalar-flat Kähler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.     

The length function of a twistor spinor with zero

We examine the Taylor expansion of the length function of a twistor spinor with zero on a Riemannian orbifold around its zero and study it on the Eguchi–Hanson orbifold. This expansion is written in some conformal normal coordinates (CNC) around the zero up to order 7. In the example of the Eguchi–Hanson orbifold, CNC are found explicitly. We use the expansion in computing the mass (a generalization of ADM–mass) of the asymptotically locally Euclidean coordinate system, which is constructed from a conformal normal coordinate system around the zero of a twistor spinor on a Riemannian spin orbifold admitting isolated singularities.     

Non-Hermitian Yang-Mills connections

We study Yang-Mills connections on holomorphic bundles over complex K?hler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk?hler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk?hler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K?hler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk?hler manifold and the space of rational curves in the twistor space of the ‘Mukai’ dual hyperk?hler manifold.     

Uniformization of small 3-orbifolds

We prove a uniformization theorem for small compact orientable 3-orbifolds, that implies Thurston's orbifold theorem.     

ALE Ricci-flat K?hler metrics and deformations of quotient surface singularities

Let N₀=\mathbb C²/H{N_0={\mathbb C}^2/H} be an isolated quotient singularity with H ì U(2){H\subset U(2)} a finite subgroup. We show that for any \mathbb Q{\mathbb Q} -Gorenstein smoothings of N ₀ a nearby fiber admits ALE Ricci-flat K?hler metrics in any K?hler class. Moreover, we generalize Kronheimer’s results on hyperk?hler 4-manifolds (J Differ Geom 29(3):685–697, 1989), by giving an explicit classification of the ALE Ricci-flat K?hler surfaces. We construct ALF Ricci-flat K?hler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat K?hler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.     

On the structure of nearly pseudo-K?hler manifolds

Firstly we give a condition to split off the K?hler factor from a nearly pseudo-K?hler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly K?hler manifolds from twistor spaces to negatively curved quaternionic K?hler manifolds and para-quaternionic K?hler manifolds. The class of nearly pseudo-K?hler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-K?hler six-manifold is Einstein.     

Oka's principle for holomorphic submersions with sprays

We classify, up to a local isometry, all non-K?hler almost K?hler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost K?hler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. Received: 1 October 2001 / Published online: 17 June 2002     

Métriques autoduales sur la boule

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-K?hler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m−1)-sphere is the boundary of a quaternionic-K?hler metric on the (4m)-ball. Oblatum 29-XI-2000 & 7-XI-2001?Published online: 1 February 2002     

On Hermitian surfaces with J-invariant Ricci tensor

We prove that a compact Hermitian surface with J-invariant Ricci tensor is K?hler provided that the difference of its scalar and conformal scalar curvature is constant. In particular, there are no locally homogeneous examples of such surfaces with odd first Betti number. Received 20 July 2000.     

Ricci Deformation of the Metric on Riemannian Orbifolds

In this note we generalize the Huisken’s (J Diff Geom 21:47–62, 1985) result to Riemannian orbifolds. We show that on any n-dimensional (n ≥ 4) orbifold of positive scalar curvature the metric can be deformed into a metric of constant positive curvature, provided the norm of the Weyl conformal curvature tensor and the norm of the traceless Ricci tensor are not large compared to the scalar curvature at each point, and therefore generalize 3-orbifolds result proved by Hamilton [Three- orbifolds with positive Ricci curvature. In: Cao HD, Chow B, Chu SC, Yau ST (eds) Collected Papers on Ricci Flow, Internat. Press, Somerville, 2003] to n-orbifolds (n ≥ 4).     

Homogeneous nearly K?hler manifolds

The structure of nearly K?hler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233?C248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly K?hler manifold is locally a Riemannian product of homogeneous nearly K?hler spaces, twistor spaces over quaternionic K?hler manifolds and six-dimensional (6D) nearly K?hler manifolds, where the homogeneous nearly K?hler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly K?hler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly K?hler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.     

Torus symmetry of compact self-dual manifolds

We classify compact anti-self-dual Hermitian surfaces and compact four-dimensional conformally flat manifolds for which the group of orientation preserving conformal transformations contains a two-dimensional torus. As a corollary, we derive a topological classification of compact self-dual manifolds for which the group of conformal transformations contains a two-dimensional torus.Partially supported by the National Science Foundation grant DMS-9306950.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler.     

A new construction of anti-self-dual four-manifolds

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D ₂ V = 0 on a five-dimensional manifold Y{\mathbf{Y}}. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in {V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space Z{\mathcal{Z}} has dim|-\frac12K_Z| 3 2{ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}, projecting thus over \mathbb C\mathbb P₂{\mathbb C\mathbb P_2} with twistor lines mapping onto plane conics.