scholarly journals EXCISION FOR SIMPLICIAL SHEAVES ON THE STEIN SITE AND GROMOV'S OKA PRINCIPLE

2003 ◽  
Vol 14 (02) ◽  
pp. 191-209 ◽  
Author(s):  
FINNUR LÁRUSSON
Keyword(s):  
Complex Manifold ◽  
Stein Manifold ◽  
Weak Equivalence ◽  
Complex Manifolds ◽  
Compact Open Topology ◽  
Homotopical Algebra ◽  
Stein Manifolds ◽  
Oka Principle ◽  
Continuous Maps ◽  
Simplicial Sheaves

A complex manifold X is said to satisfy the Oka–Grauert property if the inclusion [Formula: see text] is a weak equivalence for every Stein manifold S, where the spaces of holomorphic and continuous maps from S to X are given the compact-open topology. Gromov's Oka principle states that if X has a spray, then it has the Oka–Grauert property. The purpose of this paper is to investigate the Oka–Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka–Grauert property is equivalent to X representing a finite homotopy sheaf on the Stein site. This expresses the Oka–Grauert property in purely holomorphic terms, without reference to continuous maps.

scholarly journals Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

2006 ◽  
Vol 13 (1) ◽  
pp. 7-10
Author(s):  
Edoardo Ballico
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Open Neighborhood ◽  
Holomorphic Vector Bundle ◽  
Stein Manifold ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles

Abstract Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂(𝑈).

COMPLEXIFICATIONS OF HOLOMORPHIC ACTIONS AND THE BERGMAN METRIC

2004 ◽  
Vol 15 (08) ◽  
pp. 735-747 ◽  
Author(s):  
ANDREA IANNUZZI ◽  
ANDREA SPIRO ◽  
STEFANO TRAPANI
Keyword(s):  
Lie Groups ◽  
Complex Manifold ◽  
Analogous Result ◽  
Stein Manifold ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bergman Metric ◽  
Stein Manifolds ◽  
Invariant Domain ◽  
Necessary And Sufficient

Let G=(ℝ,+) act by biholomorphisms on a Stein manifold X which admits the Bergman metric. We show that X can be regarded as a G-invariant domain in a "universal" complex manifold X* on which the complexification [Formula: see text] of G acts. The analogous result holds for actions of a larger class of real Lie groups containing, e.g. abelian and certain nilpotent ones. For holomorphic actions of such groups on Stein manifolds, necessary and sufficient conditions for the existence of X* are given.

Decomposable Free Loop Spaces

1987 ◽  
Vol 39 (4) ◽  
pp. 938-955 ◽  
Author(s):  
J. Aguadé
Keyword(s):  
Topological Group ◽  
Loop Space ◽  
Base Point ◽  
Point Of View ◽  
Homotopy Equivalence ◽  
Compact Open Topology ◽  
Loop Spaces ◽  
Continuous Maps

In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

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.

BERGMAN AND BALANCED METRICS ON COMPLEX MANIFOLDS

2005 ◽  
Vol 02 (04) ◽  
pp. 553-561 ◽  
Author(s):  
ANDREA LOI
Keyword(s):  
Complex Manifold ◽  
Sufficient Conditions ◽  
Einstein Metric ◽  
Complex Manifolds ◽  
Balanced Metrics ◽  
Bochner’S Coordinates

In this paper we find sufficient conditions for a Bergman Einstein metric on a complex manifold to be balanced in terms of its Bochner's coordinates.

scholarly journals Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations. I

1976 ◽  
Vol 62 ◽  
pp. 55-96 ◽  
Author(s):  
Keizo Yamaguchi
Keyword(s):  
Complex Manifold ◽  
Real Hypersurface ◽  
Levi Form ◽  
Real Hypersurfaces ◽  
Conformal Transformations ◽  
Complex Manifolds ◽  
Real Analytic ◽  
Large Groups

Let S (resp. S′) be a (real) hypersurface (i.e. a real analytic sub-manifold of codimension 1) of an n-dimensional complex manifold M (resp. M′). A homeomorphism f of S onto S′ is called a pseudo-conformal homeomorphism if it can be extended to a holomorphic homeomorphism of a neighborhood of S in M onto a neighborhood of S′ in M. In case such an f exists, we say that S and S′ are pseudo-conformally equivalent. A hypersurface S is called non-degenerate (index r) if its Levi-form is non-degenerate (and its index is equal to r) at each point of S.

STRATIFIED TRANSVERSALITY OF HOLOMORPHIC MAPS

2013 ◽  
Vol 24 (13) ◽  
pp. 1350106 ◽  
Author(s):  
SAURABH TRIVEDI
Keyword(s):  
Weak Topology ◽  
Stein Manifold ◽  
Real Case ◽  
Holomorphic Maps ◽  
Countable Collection ◽  
Stein Manifolds ◽  
The Real ◽  
Oka Manifold ◽  
The Stability ◽  
Necessary And Sufficient

We discuss genericity and stability of transversality of holomorphic maps to complex analytic stratifications. We prove that the set of maps between Stein manifolds and Oka manifolds transverse to a countable collection of submanifolds in the target is dense in the space of holomorphic maps with the weak topology. This greatly generalizes earlier results on the genericity of transverse maps by Forstnerič and by Kaliman and Zaidenberg. As an application we show that the Whitney (a)-regularity of a complex analytic stratification is necessary and sufficient for the stability of transverse holomorphic maps between a Stein manifold and an Oka manifold. This gives an analogue of a theorem in the real case due to Trotman.

scholarly journals Dolbeault and J-Invariant Cohomologies on Almost Complex Manifolds

2021 ◽  
Vol 15 (7) ◽  
Author(s):  
Lorenzo Sillari ◽  
Adriano Tomassini
Keyword(s):  
Complex Manifold ◽  
Sufficient Condition ◽  
Complex Manifolds ◽  
Necessary And Sufficient Condition ◽  
Dolbeault Cohomology ◽  
Almost Complex Manifolds ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Necessary And Sufficient

AbstractIn this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.

scholarly journals Parametric jet interpolation for Stein manifolds with the density property

2019 ◽  
Vol 30 (08) ◽  
pp. 1950046
Author(s):  
Alexandre Ramos-Peon ◽  
Riccardo Ugolini
Keyword(s):  
Finite Number ◽  
Stein Manifold ◽  
Density Property ◽  
Stein Manifolds ◽  
Topological Condition ◽  
Holomorphic Flexibility

Given a Stein manifold with the density property, we show that under a suitable topological condition it is possible to prescribe derivatives at a finite number of points to automorphisms depending holomorphically on a Stein parameter. This is an Oka property of the manifold and is related to its holomorphic flexibility.

scholarly journals APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS

2013 ◽  
Vol 24 (14) ◽  
pp. 1350108 ◽  
Author(s):  
KRIS STOPAR
Keyword(s):  
Complex Manifold ◽  
Convex Domain ◽  
Holomorphic Mappings ◽  
Holomorphic Maps ◽  
Convex Domains ◽  
Exceptional Set ◽  
Proper Holomorphic Maps ◽  
Pseudoconvex Boundary ◽  
Oka Principle ◽  
Additional Assumptions

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

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