Real analytic varieties with the finiteness property and complex abelian integrals

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 .

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MILNOR FIBRATIONS AND d-REGULARITY FOR REAL ANALYTIC SINGULARITIES

International Journal of Mathematics ◽

10.1142/s0129167x10006124 ◽

2010 ◽

Vol 21 (04) ◽

pp. 419-434 ◽

Cited By ~ 10

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.

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Tangent Cones to Real Analytic Varieties

Indiana University Mathematics Journal ◽

10.1512/iumj.1973.22.22031 ◽

1972 ◽

Vol 22 (4) ◽

pp. 379-391

Author(s):

Keith Kendig

Keyword(s):

Tangent Cones ◽

Analytic Varieties ◽

Real Analytic

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Finiteness property for generalized abelian integrals

Annales de l’institut Fourier ◽

10.5802/aif.1959 ◽

2003 ◽

Vol 53 (3) ◽

pp. 767-785 ◽

Cited By ~ 1

Author(s):

Rémi Soufflet

Keyword(s):

Abelian Integrals ◽

Finiteness Property

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Sheets of Real Analytic Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-006-3 ◽

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.

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