Discreteness For the Set of Complex Structures On a Real Variety

2003 ◽  
Vol 46 (3) ◽  
pp. 321-322
Author(s):  
E. Ballico
Keyword(s):  
Complex Space ◽  
Complex Conjugate ◽  
Complex Structures ◽  
Complex Spaces ◽  
Isomorphism Classes ◽  
Real Variety ◽  
Compact Complex Space

AbstractLet X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X × Y ≅ X × W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X × Xσ ≅ W × Wσ (where Aσ denotes the complex conjugate of any variety A) is at most countable.

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 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 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.

scholarly journals Classifying affine line bundles on a compact complex space

2019 ◽  
Vol 6 (1) ◽  
pp. 103-117
Author(s):  
Valentin Plechinger
Keyword(s):  
Line Bundle ◽  
Deformation Theory ◽  
Complex Space ◽  
Main Idea ◽  
Difficult Problem ◽  
Line Bundles ◽  
Compact Complex Space ◽  
Affine Line ◽  
Picard Functor

AbstractThe classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let be a compact complex space with . We introduce the affine Picard functor which assigns to a complex space the set of families of linearly -framed affine line bundles on parameterized by . Our main result states that the functor is representable if and only if the map is constant. If this is the case, the space which represents this functor is a linear space over whose underlying set is , where is a Poincaré line bundle normalized at . The main idea idea of the proof is to compare the representability of to the representability of a functor considered by Bingener related to the deformation theory of -cohomology classes. Our arguments show in particular that, for = 1, the converse of Bingener’s representability criterion holds

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