Homogeneous Complex Manifolds Research Articles

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

On the nonexistence of S6 type complex threefolds in any compact homogeneous complex manifolds with the compact lie group G2 as the base manifold

On the nonexistence of S6 type complex threefolds in any compact homogeneous complex manifolds with the compact lie group G2 as the base manifold

Hermitian curvature flow on complex homogeneous manifolds

In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nilpotent or solvable complex Lie group equipped with an induced metric we investigate the blow-up behavior of the HCF.

Hermitian Curvature Flow on Compact Homogeneous Spaces

We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite extinction time $$T>0$$ and we analyze its behavior when $$t\rightarrow T$$ . We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.

Strata Hasse invariants, Hecke algebras and Galois representations

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks mathop {Ghbox {-}{} mathtt{Zip}}nolimits ^{mu }. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl–Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags mathop {Ghbox {-}{} mathtt{ZipFlag}}nolimits ^mu , fibered in flag varieties over mathop {Ghbox {-}{} mathtt{Zip}}nolimits ^{mu }. It provides a simultaneous generalization of the “classical case” homogeneous complex manifolds studied by Griffiths–Schmid and the “flag space” for Siegel varieties studied by Ekedahl–van der Geer. Four applications are obtained: (1) Pseudo-representations are attached to the coherent cohomology of Hodge-type Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with non-degenerate limit of discrete series Archimedean component. (3) It is shown that all Ekedahl–Oort strata in the minimal compactification of a Hodge-type Shimura variety are affine, thereby proving a conjecture of Oort. (4) Part of Serre’s letter to Tate on mod p modular forms is generalized to general Hodge-type Shimura varieties.

Complex Geometry of Iwasawa Manifolds

Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.

SPIN STRUCTURES ON COMPACT HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

Classification of Kaehler homogeneous manifolds of non-compact dimension two

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup such that $X := G/H$ is Kaehler and the codimension of the top non-vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $X$ is biholomorphic to a complex homogeneous manifold constructed using well-known basic building blocks, i.e., $\mathbb C, \mathbb C^*$, Cousin groups, and flag manifolds.

A Note on the Classification of Compact Homogeneous Locally Conformal Kähler Manifolds

In this study, we apply a result of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give some more homogeneous space involved proofs of recent classification of compact complex homogeneous locally conformal Kahler manifolds. In particular, we prove that the semisimple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semisimple Lie group S. We also prove that as an one dimensional complex torus bundle, the metrics on the manifold is completely determined by the metrics (which is the same as the Kahler class) on the base complex manifold and the metrics (same as the Kahler class) on the complex one dimensional torus.

Quaternionic-like manifolds and homogeneous twistor spaces.

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the 'quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the ρ-quaternionic manifolds. Moreover, the notion of 'heaven space' finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a ρ-quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.

On the homotopy structure of compact complex homogeneous manifolds

We consider compact complex homogeneous manifolds up to finite coverings. We give sufficient conditions under which the natural bundle for such a manifold is homotopically trivial. This triviality always holds in the case when the stationary subgroup is discrete.

Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $\pi \colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.

Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two

AbstractIf X is a connected complex manifold with dX = 2 that admits a (connected) Lie group G acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification Ĝ of G on X except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of X.

Toward a classification of real compact solvmanifolds with real symplectic structures

In this paper, we deal with the problem of classifying compact complex solvmanifolds with holomorphic symplectic structures and obtain some structure results which make the classification possible. In particular, we reduce the classification to the nilpotent case with the same dimension, which we call the nilpotent reduction. The same method also works for the real compact solvmanifolds with real symplectic structures. The real six-dimensional case was treated completely. This is one of the major steps to obtain further examples of compact holomorphic symplectic manifolds. For example, the Kodaira–Thurston surface is NOT a complex homogeneous manifold with a transitive Lie group action which keeps the complex structure invariant, but a real solvmanifold with a complex structure.

Pseudoconvex domains spread over complex homogeneous manifolds

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.

On nondegenerate coupling forms

The aim of the present paper is to investigate new classes of symplectically fat fibre bundles. We prove a general existence theorem for fat vectors with respect to the canonical invariant connections. Based on this result we give new proofs of some constructions of symplectic structures. This includes twistor bundles and locally homogeneous complex manifolds. The proofs are conceptually simpler and allow for obtaining more general results.

Classification of compact complex homogeneous manifolds with pseudo-kählerian structures

In this paper, we apply a modification theorem for a compact homogeneous solvmanifold to compact complex homogeneous manifolds with pseudo-kählerian structures. We are able to classify these manifolds as certain products of projective rational homogeneous spaces, tori, simple and double reduced primitive pseudo-kähler spaces.

Positive toric fibrations

A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibred over M/T, with fibre T. We discuss the notion of positivity in fibre bundles and define positive toric bundles. Given an irreducible complex subvariety X ⊂ M of a positive principal toric bundle, we show that either X is T-invariant, or it lies in an orbit of a T-action. For principal elliptic bundles, this theorem is known from Verbitsky [Math. Res. Lett. 12 (2005) 251–264]. As follows from the Borel–Remmert–Tits theorem, any simply connected compact homogeneous complex manifold is a principal toric bundle. We show that compact Lie groups with left-invariant complex structure I are positive toric bundles, if I is generic. Other examples of positive toric bundles are discussed.

A Levi problem for discontinuous strictly q -plurisubharmonic functions

In this article we consider a Levi problem for discontinuous strictly q-plurisubharmonic functions on complex spaces that admit a stratification by infinitesimally homogeneous complex manifolds.

Modification and the cohomology groups of compact solvmanifolds

In this note we give a modification theorem for a compact homogeneous solvmanifold such that a certain Mostow type condition will be satisfied. An application of this result is a simpler way to calculate the cohomology groups of compact quotients of real solvable Lie group over a cocompact discrete subgroup. Furthermore, we apply the second result to obtain a splitting theorem for compact complex homogeneous manifolds with symplectic structures. In particular, we are able to classify compact complex homogeneous spaces with pseudo-Kählerian structures.