scholarly journals A smoothness criterion for complex spaces in terms of differential forms

2020 ◽  
Vol 126 (2) ◽  
pp. 221-228
Author(s):  
Håkan Samuelsson Kalm ◽  
Martin Sera
Keyword(s):  
Complex Space ◽  
Differential Forms ◽  
Complex Spaces ◽  
Smoothness Criterion

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha _X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $\omega _X^1$.

scholarly journals Reflexive differential forms on singular spaces. Geometry and cohomology

2014 ◽  
Vol 2014 (697) ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Thomas Peternell
Keyword(s):  
Differential Forms ◽  
Extension Theorem ◽  
Singular Variety ◽  
Singular Spaces ◽  
Complex Spaces ◽  
Rationally Connected ◽  
Smooth Locus ◽  
Recent Extension ◽  
Rationally Connected Varieties

AbstractBased on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials.First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

scholarly journals On The Holomorphic Automorphism Groups of Complex Spaces

1968 ◽  
Vol 33 ◽  
pp. 85-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Lie Group ◽  
Complex Space ◽  
Automorphism Groups ◽  
Analogous Result ◽  
Holomorphic Mapping ◽  
Compact Complex Manifold ◽  
Holomorphic Automorphism ◽  
Complex Spaces ◽  
Weaker Conditions ◽  
Holomorphic Automorphism Groups

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

scholarly journals On the Automorphism Group of a Holomorphic Fiber Bundle over A Complex Space

1970 ◽  
Vol 37 ◽  
pp. 91-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Fiber Bundle ◽  
Complex Space ◽  
Automorphism Groups ◽  
Compact Complex Manifold ◽  
Compact Open Topology ◽  
Complex Spaces ◽  
Holomorphic Fiber Bundle ◽  
Holomorphic Automorphisms

In [8], A. Morimoto proved that the automorphism group of a holomorphic principal fiber bundle over a compact complex manifold has a structure of a complex Lie group with the compact-open topology. The purpose of this paper is to get similar results on the automorphism groups of more general types of locally trivial fiber spaces over complex spaces. We study automorphisms of a holomorphic fiber bundle over a complex space which has a complex space Y as the fiber and a (not necessarily complex Lie) group G of holomorphic automorphisms of Y as the structure group (see Definition 3. l).

scholarly journals On bimeromorphic automorphisms of hyperbolic complex spaces

1979 ◽  
Vol 73 ◽  
pp. 1-5 ◽  
Author(s):  
Akio Kodama
Keyword(s):  
Complex Space ◽  
Complex Spaces ◽  
Hyperbolic Complex

Let X be a hyperbolic complex space in the sense of S. Kobayashi [2]. We write Aut(X)(resp. Bim (X)) for the group of all biholomorphic (resp. bimeromorphic) automorphisms of X.

scholarly journals Symplectic cutting of Kähler manifolds

1999 ◽  
Vol 1999 (508) ◽  
pp. 85-98
Author(s):  
Maxim Braverman
Keyword(s):  
Vector Bundle ◽  
Complex Space ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Symplectic Reduction ◽  
Kahler Manifolds ◽  
Complex Spaces ◽  
Irreducible Components ◽  
Flat Family

Abstract We obtain estimates on the character of the cohomology of an S1-equivariant holomorphic vector bundle over a Kähler manifold M in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of M. In particular, we prove and extend inequalities conjectured by Wu and Zhang. The proof is based on constructing a flat family of complex spaces Mt (t ∈ ℂ) such that Mt is isomorphic to M for t ≠ 0, while M0 is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.

A definability result for compact complex spaces

2004 ◽  
Vol 69 (1) ◽  
pp. 241-254 ◽  
Author(s):  
Dale Radin
Keyword(s):  
Algebraic Geometry ◽  
Complex Space ◽  
Analogous Result ◽  
Order Structure ◽  
Holomorphic Map ◽  
Complex Spaces ◽  
Compact Complex Space ◽  
Image Position

AbstractA compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X x X, … as basic relations. Let f: X → Y be a proper surjective holomorphic map between complex spaces and set Xy ≔ f−1(y). We show that the setis analytically constructible, i.e.. is a definable set when X and Y are compact complex spaces and f: X → Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.

scholarly journals The polarization constant of finite dimensional complex spaces is one

2021 ◽  
pp. 1-19
Author(s):  
VERÓNICA DIMANT ◽  
DANIEL GALICER ◽  
JORGE TOMÁS RODRÍGUEZ
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Homogeneous Polynomial ◽  
Complex Space ◽  
Necessary Conditions ◽  
Best Constant ◽  
The Real ◽  
Complex Spaces ◽  
Finite Dimensional ◽  
Image Position

Abstract The polarization constant of a Banach space X is defined as \[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\] where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\] . We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\] . We derive some consequences of this fact regarding the convergence of analytic functions on such spaces. The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

On nondifferentiable minimax semi-infinite programming problems in complex spaces

2016 ◽  
Vol 23 (3) ◽  
pp. 367-380
Author(s):  
Anurag Jayswal ◽  
Krishna Kummari
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Complex Space ◽  
Sufficient Optimality Conditions ◽  
Dual Problems ◽  
Complex Spaces ◽  
Infinite Programming ◽  
Primal And Dual Problems ◽  
Primal And Dual ◽  
Necessary And Sufficient

AbstractThe purpose of this paper is to study a nondifferentiable minimax semi-infinite programming problem in a complex space. For such a semi-infinite programming problem, necessary and sufficient optimality conditions are established by utilizing the invexity assumptions. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. In order to relate the primal and dual problems, we have also derived appropriate duality theorems.

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