Remarks on k-Leviflat Complex Manifolds

1979 ◽  
Vol 31 (4) ◽  
pp. 881-889 ◽  
Author(s):  
B. Gilligan ◽  
A. Huckleberry
Keyword(s):  
Plurisubharmonic Function ◽  
Stein Manifold ◽  
Levi Form ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Complex Manifolds ◽  
Generic Fiber ◽  
Convex Case ◽  
Compact Complex Manifolds

In the theory of functions of several complex variables one is naturally led to study non-compact complex manifolds which have certain types of exhaustions. For example, on a Stein manifold X there is a strictly plurisubharmonic function ϕ: X → R+ so that the pseudoballs Bc = {φ < c } exhaust X. Conversely, a manifold which has such an exhaustion is Stein. The purpose of this note is to study a class of manifolds which have exhaustions along the lines of those on holomorphically convex manifolds, namely the k-Leviflat complex manifolds. Unlike the Stein case, the Levi form may have positive dimensional 0-eigenspaces. In the holomorphically convex case these are tangent to the generic fiber of the Remmert reduction.

scholarly journals Some applications of Synge’s formula to the theory of several complex variables

1977 ◽  
Vol 67 ◽  
pp. 53-64 ◽  
Author(s):  
Takeshi Sasaki ◽  
Osamu Suzuki
Keyword(s):  
Sectional Curvature ◽  
Pseudoconvex Domain ◽  
Stein Manifold ◽  
Kähler Manifold ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Kahler Manifold

In [10] and [11], the second author proved the following theorem by using Synge’s formula:THEOREM I. Let M be a kähler manifold with positive holomorphic bi-sectional curvature. Then every pseudoconvex domain in M is a Stein manifold.

scholarly journals Schottky groups acting on homogeneous rational manifolds

2019 ◽  
Vol 2019 (753) ◽  
pp. 23-56 ◽  
Author(s):  
Christian Miebach ◽  
Karl Oeljeklaus
Keyword(s):  
Deformation Theory ◽  
Group Actions ◽  
Picard Group ◽  
Schottky Group ◽  
Fundamental Groups ◽  
Complex Manifolds ◽  
Schottky Groups ◽  
Algebraic Dimension ◽  
Geometric Invariants ◽  
Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

scholarly journals Higher-page Bott–Chern and Aeppli cohomologies and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dan Popovici ◽  
Jonas Stelzig ◽  
Luis Ugarte
Keyword(s):  
Positive Integer ◽  
Spectral Sequence ◽  
Complex Manifolds ◽  
Serre Duality ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

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