Slicing and Intersection Theory for Chains Modulo ν Associated With Real Analytic Varieties

1973 ◽  
Vol 183 ◽  
pp. 327 ◽  
Author(s):  
Robert M. Hardt
Keyword(s):  
Intersection Theory ◽  
Analytic Varieties ◽  
Real Analytic

Embeddings of Real Analytic Varieties or Spaces

1986 ◽  
pp. 109-128
Author(s):  
Francesco Guaraldo ◽  
Patrizia Macrì ◽  
Alessandro Tancredi
Keyword(s):  
Analytic Varieties ◽  
Real Analytic

Real Analytic Varieties

1986 ◽  
pp. 60-82
Author(s):  
Francesco Guaraldo ◽  
Patrizia Macrì ◽  
Alessandro Tancredi
Keyword(s):  
Analytic Varieties ◽  
Real Analytic

Higher Order Tangents to Analytic Varieties along Curves

2003 ◽  
Vol 55 (1) ◽  
pp. 64-90 ◽  
Author(s):  
Rüdiger W. Braun ◽  
Reinhold Meise ◽  
B. A. Taylor
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Forthcoming Paper ◽  
Puiseux Series ◽  
Open Set ◽  
Partial Differential Operators ◽  
Real Analytic Functions ◽  
Pure Dimension ◽  
Analytic Varieties ◽  
Real Analytic

AbstractLet V be an analytic variety in some open set in which contains the origin and which is purely k-dimensional. For a curve γ in , defined by a convergent Puiseux series and satisfying γ(0) = 0, and d ≥ 1, define Vt := t−d(V − (t)). Then the currents defined by Vt converge to a limit current Tγ,d[V] as t tends to zero. Tγ,d[V] is either zero or its support is an algebraic variety of pure dimension k in . Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on .

MILNOR FIBRATIONS AND d-REGULARITY FOR REAL ANALYTIC SINGULARITIES

2010 ◽  
Vol 21 (04) ◽  
pp. 419-434 ◽  
Author(s):  
J. L. CISNEROS-MOLINA ◽  
J. SEADE ◽  
J. SNOUSSI
Keyword(s):  
Fiber Bundle ◽  
Critical Value ◽  
Singular Set ◽  
Small Sphere ◽  
Milnor Fibration ◽  
Analytic Varieties ◽  
Real Analytic ◽  
Analytic Singularities ◽  
Analytic Maps ◽  
Do So

We study Milnor fibrations of real analytic maps [Formula: see text], n ≥ p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f-1(0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom af-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/‖f‖. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in ℝn the spherefication gives a fiber bundle equivalent to the Milnor fibration.

Sheets of Real Analytic Varieties

1960 ◽  
Vol 12 ◽  
pp. 51-67
Author(s):  
Andrew H. Wallace
Keyword(s):  
General Idea ◽  
Algebraic Varieties ◽  
Analytic Variety ◽  
Analytic Case ◽  
Real Algebraic Varieties ◽  
Analytic Varieties ◽  
Local Properties ◽  
Real Analytic ◽  
Real Analytic Variety

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.

Higher Order Tangents to Analytic Varieties along Curves. II

2008 ◽  
Vol 60 (1) ◽  
pp. 33-63 ◽  
Author(s):  
Rüdiger W. Braun ◽  
Reinhold Meise ◽  
B. A. Taylor
Keyword(s):  
Finite Number ◽  
Algebraic Variety ◽  
Higher Order ◽  
Algebraic Varieties ◽  
Analytic Variety ◽  
Open Set ◽  
Local Case ◽  
Analytic Varieties ◽  
Real Analytic ◽  
Analytic Curves

AbstractLet V be an analytic variety in some open set in ℂn. For a real analytic curve γ with γ(0) = 0 and d ≥ 1, define Vt = t−d(V − γ(t)). It was shown in a previous paper that the currents of integration over Vt converge to a limit current whose support Tγ,δV is an algebraic variety as t tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the Vt. As a corollary, it is shown that Tγ,δV is either inhomogeneous or coincides with Tγ,δV for all δ in some neighborhood of d. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragmén-Lindelöf conditions. Corresponding results for limit varieties Tσ,δW of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.

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