A Classification of Homogeneous Surfaces

Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

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 purpose of the present note is to show that holomorphic fiber bundles with hyperbolic fiber are almost trivial : They are trivial if the base is simply connected, and have a simple structure otherwise. Kobayashi [2],[3] has introduced the notion of a hyperbolic complex manifold.We need not be concerned with the definition here, but only with the following consequence [3], [4]: If M is a hyperbolic complex manifold, p a point of M, and U a neighborhood of /, then there is a <5>0 such that any holomorphic map <f> of the unit disk into M with (f,(0)=p maps the disk {z:|z|<} into U. From this, it follows that, if iv1, , wn are coordinates at/, and / an integer, then there are constants Al depending only on / and the choice of coordinates such that, if-p is any map of the unit disk into M with cf>(0)=p, then \dlwidzl\<Al at z=0.The crux of our argument is the following proposition, which generalizes a result of Kobayashi's that a complex Lie group (of dimension greater than 0) cannot act holomorphically on a hyperbolic manifold. Proposition.Let M and N be complex manifolds with M hyperbolic.Let F=F(w, z) be a holomorphic map of MxN into M. Suppose that, for some z0 e N, the restriction of F to Mx{z0} is a biholomorphic map onto M. Then F is independent of z; i.e., F(w, z)=F(w, z0). Proof.It suffices to prove the proposition for the case that N is a ball, and hence for the case that Nis a disk A, which we take to be the unit disk with z0=0.Without loss of generality, we may assume F(w, z0)=w.Near a point (w0, z0), the map F has a power series expansion of the form F\w, z) = w' + a\w)zl + 0(zl+1).-

Bergman kernels related to Hermitian line bundles over compact complex manifolds by B. Berndtsson A gentle introduction to points of finite type on real hypersurfaces by J. P. D'Angelo The moduli space of 2-step nilpotent Lie algebras of type $(p,q)$ by P. Eberlein The homotopy principle in complex analysis: A survey by F. Forstneric Finiteness theorems in riemannian geometry by K. Grove Generalized Toponogov's theorem for manifolds with radial curvature bounded below by Y. Itokawa, Y. Machigashira, and K. Shiohama The global isometric embedding problem by H. Jacobowitz The Bergman metric invariants and their boundary behavior by K.-T. Kim and S. G. Krantz Natural connections in almost complex manifolds by S. Kobayashi The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type by S. Kumar, B. Leeb, and J. Millson Subelliptic estimates and scaling in the $\bar\partial$-Neumann problem by J. D. McNeal Negativity of curvature on spaces parametrizing Hodge decompositions of reduced first cohomology groups by N. Mok On the extension of $L^2$ holomorphic functions VI--A limiting case by T. Ohsawa Variations on a theme of Synge by P. Petersen Mappings between real submanifolds in complex space by L. P. Rothschild Harmonic analysis on toric varieties by B. Shiffman, T. Tate, and S. Zelditch On complex manifolds with noncompact automorphism groups by B. Wong Kahler manifolds with slightly positive bisectional curvature by H.-H. Wu and F. Zheng.

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.

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Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

Consider a holomorphic vector bundle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> and an open cover <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝔘</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo lspace="0pt">:</mml:mo> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , parametrized by a complex manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . We prove that the sheaf cohomology groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> can be computed from the complex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>hol</mml:mi> <mml:mo>•</mml:mo> </mml:msubsup> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>𝔘</mml:mi> <mml:mo>,</mml:mo> <mml:mi>L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of cochains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>...</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> that depend holomorphically on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> , provided <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> <mml:mo>×</mml:mo> <mml:mi>X</mml:mi> <mml:mo lspace="0pt">:</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a Stein open subset of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>×</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> . The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomorphic action of a complex reductive Lie group on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> .

We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we generalize Leray-Hirsch, K\"{u}nneth and Poincar\'{e}-Serre duality theorems on them. At last, a blowup formula is given, which affirmatively answers a question posed by Chen, Y. and Yang, S. in \cite{CY}.

: Stability of compact submanifolds of complex manifolds and some related topics are discussed. A compact submanifold V of a complex manifold W is said to be stable if any small deformation W T OF contains a small deformation Vt of V. Let psi be the sheaf over V of germs of holomorphic sections of the normal bundle of V in W. If the first cohomology group H1(V,psi) vanishes then V is a stable submanifold of W. A fibre structure of a compact fibred complex manifold M is said to be stable if any small deformation Mt of M retains a fibre structure. If each fibre of M is regular then the fibre structure of M is stable. (Author)

Let ι : C2 ↪→ S be a compactification of the two dimensional complex space C2. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 ([2]) that S is a rational surface. Here we deal with a more general meromorphic map f : Cn → X into a compact complex manifold X of dimension n, whose differential df has generically rank n. Let ρf denote the order of f . We will prove that if ρf < 2, then every global symmetric holomorphic tensor must vanish; in particular, if dimX = 2 and X is kahler, then X is a rational surface. Without the kahler condition there is no such conclusion, as we will show by a counter-example using a Hopf surface. This may be the first instance that the kahler or non-kahler condition makes a difference in the value distribution theory. 2010 Mathematics Subject Classification. Primary 32H30; Secondary 14M20.

Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the (C*)^m-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi-Eckmann manifolds, and their deformations. We construct transversely Kaehler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kaehler submanifold (or, more generally, a Fujiki class C subvariety) in such a moment-angle manifold is contained in a leaf of the foliation F. For a generic moment-angle manifold Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that the algebraic dimension of Z is zero.

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.