Generalized Hopf Manifolds Research Articles

A new positivity condition for the curvature of Hermitian manifolds

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kahler case. We derive a Bochner formula for closed (1, 1)-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfies our positivity condition, then any class $$\alpha \in H^{1, 1}_{BC}(X)$$ can be represented by a closed (1, 1)-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

The canonical foliation of a compact generalized Hopf manifold

Let M be a compact generalized Hopf (g.H.) manifold (i.e., a locally conformal Kähler manifold with parallel Lee form) defined by I. Vaisman. We study several properties of the canonical foliation on M. The main results of this paper are the following: 1. (1) We can canonically lift the foliation to any holomorphic line bundle F on M. Moreover holomorphic sections of F are foliated one. 2. (2) There exist compact leaves. 3. (3) We describe the geometric structure of a compact g.H. manifold on which the canonical foliation is regular.

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S1 ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.

Self-dual geometry of generalized Hermitian surfaces

Several results on the geometry of conformally semiflat Hermitian surfaces of both classical and hyperbolic types (generalized Hermitian surfaces) are obtained. Some of these results are generalizations and clarifications of already known results in this direction due to Koda, Itoh, and other authors. They reveal some unexpected beautiful connections between such classical characteristics of conformally semiflat (generalized) Hermitian surfaces as the Einstein property, the constancy of the holomorphic sectional curvature, and so on. A complete classification of compact self-dual Hermitian -surfaces that are at the same time generalized Hopf manifolds is obtained. This provides a complete solution of the Chen problem in this class of Hermitian surfaces.

Compact generalized Hopf and cosymplectic solvmanifolds and the Heisenberg groupH(n, 1)

In this paper we obtain a generalized Hopf structure on the total space of certain principal circle bundles over a compact cosympletic manifold. Using this result we give new examples of compact generalized Hopf manifolds. We describe these examples as suspensions with fibre a compact quotient of the generalized Heisenberg groupH(n, 1) by a discrete subgroup and we show an explicit realization of them as compact solvmanifolds.

Holomorphic Maps of Compact Generalized Hopf Manifolds

We investigate holomorphic maps between compact generalized Hopf manifolds (i.e., locally conformal Kahler manifolds with parallel Lee form). We show that they preserve the canonical foliations. Moreover, we study compact complex submanifolds of g.H. manifolds and holomorphic submersions from compact g.H. manifolds.

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, if∼M is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection of∼M is quasi-Einstein, too, provided thatM is tangent to the Lee field of∼M. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ m (of a complex Hopf manifoldS2m+1 ×S1).

Locally conformal K�hler submersions

In this paper we study almost Hermitian submersions with total space a locally conformal Kahler (l.c.K.) manifold, i.e., l.c.K. submersions. We derive necessary and sufficient conditions for the fibers of a l.c.K. submersion to be minimal and for the horizontal distribution to be completely integrable. We give, under certain conditions, some relations between the Betti numbers of the total space and the base space of a l.c.K. submersion and we obtain all the l.c.K. submersions with totally geodesic fibers and total space a particular class of generalized Hopf manifolds.

Extrinsic spheres of a generalized Hopf manifold

Extrinsic spheres of a generalized Hopf manifold

Submanifolds of generalized Hopf manifolds, type numbers and the first Chern class of the normal bundle

Any orientable real hypersurface M of a complex Hopf manifold (carrying the locally conformal Kaehler (l.c.K.) metric discovered by I.Vaisman [33]) has a natural f-structure P as a generic Cauchy-Riemann submanifold; we show (cf. our § 5) that if P anti-commutes with the Weingarten operator, then the type number of the hypersurface is less equal than 1. Moreover, M carries the natural almost contact metrical structure observed by Y.Tashiro [30]; if its almost contact vector is an eigenvector of the Weingarten operator corresponding to a nowhere vanishing eigenfunction and the holomorphic distribution is involutive, then M is foliated with globally conformai Kaehler manifolds (cf. our § 5), provided that some restrictions on the type number of M are imposed. We derive (cf. our § 6) a «Simons type» formula and apply it to compact orientable hypersurfaces with non-negative sectional curvature (in a complex Hopf manifold) and parallel mean curvature vector. Several examples of submanifolds of l.c.K. manifolds are exhibited in § 3. Our § 7 studies complex submanifolds of generalized Hopf manifolds; for instance, we show that the first Chern class of the normal bundle of a complex submanifold having a flat normal connection is vanishing.

Generalized Hopf manifolds, locally conformal Kaehler structures and real hypersurfaces

We study the geometry of submanifolds of complex Hopf manifolds endowed with the (locally conformal Kaehler) Boothby metric. 1. Generalized Hopf manifolds and the Boothby metric. Let fleC, 0 l , generated by z^az, z(=W. Then Ga acts freely and properly discontinuously on W, see [28], vol. II, p. 137, so that the quotient space Hna=W/Ga becomes in a natural way a complex w-dimensional manifold. This is the well known complex Hopf manifold. In their attempt to construct complex structures on products SxL, where S is the unit circle and L an odd dimensional homotopy sphere, E. Brieskorn & A. Van de Ven, [3], have generalized Hopf manifolds a follows. Let n > l and (fe0, ••• , bn)^Z , bj^l, 0£j£n. Let (z0, ••• , zn) be the natural complex coordinates on C n + 1 . Define X(b) = X2n(b0, ••• , fc»)cC n+1 by the equation: Then X(b) ia an aίϊine algebraic variety with one singular point at the origin of C if bj^2, / = 0 , - n (and without singularities if b3— for at least one /). Next B{b)—X(b)— {0} is a complex n-dimensional manifold, referred hereafter as the Brieskorn manifold determined by the integers b0, ••• , bn. See [2]. There is a natural holomorphic action of C on B(b) given by: t(Zo, —, ^ n ) = ( ^ O e X p ( y ^ ) , ••• , Zn ΘXp (—-y^ ) ) (1) where f e C , wa=— log \a \— iΦa, Φα=arctan(/m(α)//?β(fl)), —π/2 D x 5»(« defined by f(a, jc)=(α, ί/αx), for any a e D 1 , xe=B(b), where £/αe=GL(n+l, C) is the matrix: Note that / is an automorphism of DxBφ). The action of GL(n+l, C) on C n + 1 induces an action of Z « { / 7 m e Z } on DxBφ). Let: be the quotient space. We establish the following: THEOREM 1. X is a complex n-dimensional manifold. Moreover, if n—2, then there exists a surjective holomorphic map π: X-^D which makes X into a complex analytic family of compact complex surfaces, for any a^D there is a diffeomorphism between π~\a) and Hl(b). We recall that a triple (X, π, M) is a complex analytic family of compact complex manifolds if X, M are complex manifolds and π: X-*M is a proper holomorphic map which is of maximal rank at all points of X. Then each fibre π~\a G G M , is a compact complex manifold. Note that the action (1) of Z on Bφ) generalizes slightly the one in [3], p. 390. There B(l, ••• , 1)/Z is diffeomorphic to W/G1/e. The proof of Theorem 1 is organized in several steps, as follows. STEP 1. Z acts freely on DxB(b). Let (a, x) be a fixed point of / m , m e Z . Thus Ufx = x, and consequently: for O^tj ^n. Since at least one z3 is non-zero, it follows that m=0. STEP 2. {//meZ} is a properly discontinuous group of analytic transformations of DxBψ). Let KdD, LcB(b) be compact subsets. It is enough to show that the set of all m e Z with the property:

Generalized Hopf manifolds with flat local Kaelher metrics

Generalized Hopf manifolds with flat local Kaelher metrics

Generalized Hopf manifolds

Let (M,J,g) be a Hermitian manifold with complex structure J, metric g, and Kahler form Ω. Then g is locally conformal Kahler iff dΩ=ω ∧ Ω for some closed and non-exact 1-form ω. Moreover, if ω is a parallel form, M is called a generalized Hopf manifold. The main results of this paper are: (a) the description of the geometric structure of the compact locally conformal Kahler-flat manifolds; (b) the description of the geometric structure of the compact generalized Hopf manifolds on which a certain canonically defined foliation is regular; (c) a description of the harmonic forms and Betti numbers of a general compact generalized Hopf manifold; (d) a method for studying analytic vector fields on generalized Hopf manifolds; (e) conditions for submanifolds of generalized Hopf manifolds to belong to the same class.