Explorations in Complex and Riemannian Geometry

We compare the second fundamental forms of a family of parallel hypersurfaces in different Riemannian manifolds. This leads to new proofs for the distance and volume comparison theorems in Riemannian geometry. In particular, we get a new result on the volume of the set of points with distance≤r from a totally geodesic submanifold, for any r. The analytic prerequisite is the investigation of the Riccati type ODE which is satisfied by the second fundamental form of a parallel hypersurface family.

Let (E, h) be a semipositive hermitian holomorphic line bundle on a compact complex manifold X with dimX>11$$\\end{document}]]>. Assume that for each point x∈X there exists a divisor D∈|E| in the complete linear system determined by E whose complement X\\D is a Stein neighbourhood of x with the density property. Then, the disc bundle Δh(E)={e∈E:|e|h<1} is an Oka manifold while Dh(E)={e∈E:|e|h>1}1\\}$$\\end{document}]]> is a Kobayashi hyperbolic domain. In particular, the zero section of E admits a basis of Oka neighbourhoods {|e|h<c} with c>00$$\\end{document}]]>. We show that this holds if X is a rational homogeneous manifold of dimension >11$$\\end{document}]]>. This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.

The aim of this paper is to collect some interesting unsolved problems in the topology of compact complex manifolds which are scattered in literature; needless to remark, some of them are known to specialists. Besides, the readermay consult the well-known list of problems by F. Hirzebruch [ 5 ] which has stimulated remarkable progress in this direction. But the topology of complex manifolds has always baffled workers in this field and is still far from satisfactory. Of course, there is remarkable progress in the case of homogeneous compact complex manifolds. We start with the following result.

In the first half of the twentieth century, new tools were developed for the study of the algebraic topology of manifolds, and in particular, complex manifolds: de Rham’s representation of algebraic topology with differential forms, Kahler’s introduction of specific types of Riemannian metrics for complex manifolds, and Hodge’s theory of harmonic differential forms. These tools, along with the theory of sheaf cohomology developed by Leray somewhat later, allowed Kodaira to formulate and prove in 1954 an embedding theorem for compact complex manifolds which completely characterized the abstract manifolds that could be embedded as a submanifold of a complex projective space.

A real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ℙ2.

This is a survey dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case. We investigate holomorphic foliations with a transverse holomorphic Cartan geometry, and also with the more general structure of branched transverse holomorphic Cartan geometry.The first part of the chapter presents the geometric notion of holomorphic G-structure whose origin was motivated by various classical examples. We explain some classification results for compact complex manifolds endowed with holomorphic G-structures highlighting the special case of \(\mathrm {GL}(2,{\mathbb {C}})\)-structures and \(\mathrm {SL}(2,{\mathbb {C}})\)-structures. The second part of the survey deals with holomorphic Cartan geometry in the classical case and in the branched and generalized case. Two definitions of foliated (branched, generalized) Cartan geometry are described and shown to be equivalent. We provide some classification results of compact complex manifolds with foliated (branched or generalized) Cartan geometries. At the end some related open problems are formulated.KeywordsHolomorphic geometric structureHolomorphic GL(2)-structureKähler manifoldFoliated principal bundleFoliated Cartan geometry1991 Mathematics Subject Classification53C0753C1032Q57

The main result can be stated roughly as follows: Let M M be an Alexandrov space, Ω ⊂ M \Omega \subset M an open domain and f : Ω → R f:\Omega \to \mathbb {R} a harmonic function. Then f f is Lipschitz on any compact subset of Ω \Omega . Using this result I extend proofs of some classical theorems in Riemannian geometry to Alexandrov spaces.

Deformations of Levi flat hypersurfaces in complex manifolds

We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.

The compact complex manifolds considered in this article are principal torus bundles over a torus. We consider the Kodaira Spencer map of the complete Appell Humbert family (introduced by the first author in Part I) and are able to show that we obtain in this way a connected compo- nent of the space of complex structures each time that the base dimension is two, the fibre dimension is one, and a suitable topological condition is verified.

We consider various types of special Hermitian metrics on compact complex manifolds, namely Kahler, Hermitian-symplectic, SKT, balanced and strongly Gauduchon metrics. We study the strongly Gauduchon metrics in relation to the SKT condition on complex nilmanifolds and to the Hermitian-symplectic condition on certain complex solvmanifolds. We also review the behaviour of the existence properties of these special metrics on compact complex manifolds under holomorphic deformations.

The famous Laplace-Beltrami operator △ acting on differential forms on a Riemannian manifold M determines in some sense the geometry of M. For example the Hodge decomposition theorem implies that in the compact case $$ X(M) = Trace {e^{{ - t{\Delta_{{even}}}}}} - Trace {e^{{ - t{\Delta_{{odd}}}}}} $$ where X(M) is the Euler characteristic of M and \( {\Delta_{\text{even}}} = \Delta {\left| {_{p{even}}^{ \oplus }\Lambda P(M),{ }{\Delta_{\text{odd}}} = \Delta } \right|_p}_{\text{odd}}^{ \oplus }\Lambda P \) Of course the theory of the operator in the complex case is much richer. We are going to give a short review of the theory of the Laplace-Beltrami operator on compact complex manifolds. In particular, the Hodge decomposition and its applications will be given. The case of a compact Kahler manifold will also be mentioned. Some other elliptic operators essentially connected with the geometry of M will be introduced. One of them is the so-called Ahlfors-Laplacian S*S acting on 1-forms. S is the Ahlfors’ operator which arises naturally in the theory of quasiconformal deformations of M. S*S is strongly influenced by the geometry of M. It behaves nicely both in the real and in the complex cases. Before passing to the operators some necessary information from the theory of real and complex geometry will be given.

Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.

We prove a classification theorem by cohomology classes for compact Riemannian manifolds with a one-parameter group of isometries without fixed points generalizing the classification of line bundles (more precisely, their circle bundles) over compact manifolds by their first Chern class. We also prove a classification theorem generalizing that of holomorphic line bundles over compact complex manifold by the Picard group of the base for a subfamily of manifolds with additional structure resembling that of circle bundles of such holomorphic line bundles.

We describe the norming sets for the space of global holomorphic sections to a k-power of a positive holomorphic line bundle on a compact complex manifold X. We characterize in metric terms the sequence of measurable subsets $$\{G_{k}\}_{k}$$ of X such that there is a constant $$C > 0$$ where $$\begin{aligned} \Vert s\Vert ^{2}\le C \int _{G_{k}} |s(z)|^{2} \mathrm{{d}}V(z) \end{aligned}$$ for every $$s\in H^{0}(X,{\mathcal {O}}(L^{\otimes k}))$$ and for all $$k\in {\mathbb {N}}$$ .