This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G C /P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kähler-Ricci solitonic metric if and only if it is Fano.2000 Mathematics Subject Classification. 14M15, 32M12, 53C55.
Let M be a CR manifold of hypersurface type, which is Levi degenerate but also satisfying a k-nondegeneracy condition at all points. This might be only if dim M ≥ 5 and if dim M = 5, then k = 2 at all points. We prove that for any 5-dimensional, uniformly 2-nondegenerate CR manifold M there exists a canonical Cartan connection, modelled on a suitable projective completion of the tube over the future light cone {z ∈ C 3 : (x 1 ) 2 +(x 2 ) 2 −(x 3 ) 2 = 0 , x 3 > 0}. This determines a complete solution to the equivalence problem for this class of CR manifolds.2010 Mathematics Subject Classification. 32V05, 32V40, 53C10.
A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point xo, so that: a) it is a smooth solution on M \{xo} to the Monge-Ampère equation (dd c u) n = 0; b) xo is a singular point for u of logarithmic type and e u extends smoothly on the blow up of M at xo; c) dd c (e u ) > 0 at any point of M \ {xo}. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of C n .The moduli spaces of bounded manifolds of circular type are studied. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n 2 when M is a (nonconvex) circular domain while it is of dimension n 2 + 2n when M is a strictly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.2000 Mathematics Subject Classification. 32G05, 32W20, 32Q45. Key words and phrases. Manifolds of circular type, Monge-Ampère equations, strictly convex domains, deformations of complex structures. 1 iv) dd c τ o (φ(X), φ(X)) < dd c τ o (X, X) for any 0 = X ∈ H 0,1 .Proof. First of all, recall that by the J o -invariance of the 2-form dd c τ o , for any two vector fields in H 0,1 ⊕ Z 0,1 or in H 1,0 ⊕ Z 1,0 ,So, from the proof of Theorem 4.2, the reader can check that a complex structure J of Lempert type is so that (B n , J, τ o ) is a manifold of circular type, if and only if for any 0 = X ∈ H 0,1This proves (iv). For checking the necessity and sufficiency of (i) -(iii), we only need to show that those properties are necessary and sufficient condition for the integrability of the unique almost complex structure J, which coincides with J o on the radial discs, leaves the distribution H invariant and have an associated antiholomorphic distribution H 0,1 J which is as in (5.2). Such almost complex structure J is integrable if and only if for any anti-holomorphic vector fields X, Y ∈ H 0,1 one has[Z 0,1 , X + φ(X)] ∈ Z 0,1 + H 0,1 J
Abstract. We consider 6-dimensional strict nearly Kähler manifolds acted on by a compact, cohomogeneity one automorphism group G. We classify the compact manifolds of this class up to G-diffeomorphisms. We also prove that the manifold has constant sectional curvature whenever the group G is simple.
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