scholarly journals On singular sets of flat holomorphic mappings with isolated singularities

1978 ◽  
Vol 70 ◽  
pp. 47-80
Author(s):  
Hideo Omoto
Keyword(s):  
Algebraic Geometry ◽  
Singular Point ◽  
Critical Points ◽  
Euler Number ◽  
Holomorphic Mappings ◽  
Algebraic Varieties ◽  
Isolated Singularities ◽  
General Fiber ◽  
Singular Sets ◽  
Algebraic Maps

In [4] B. Iversen studied critical points of algebraic mappings, using algebraic-geometry methods. In particular when algebraic maps have only isolated singularities, he shows the following relation; Let V and S be compact connected non-singular algebraic varieties of dimcV = n, and dimc S = 1, respectively. Suppose f is an algebraic map of V onto S with isolated singularities. Then it follows thatwhere χ denotes the Euler number, μf(p) is the Milnor number of f at the singular point p, and F is the general fiber of f : V → S.

scholarly journals On the Differential Forms on Algebraic Varieties

1952 ◽  
Vol 4 ◽  
pp. 73-78
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Algebraic Geometry ◽  
Differential Form ◽  
Differential Forms ◽  
Multiple Point ◽  
Algebraic Varieties ◽  
Complete Variety ◽  
Case Characteristic

In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).

On the removability of isolated singular points for elliptic equations involving variable exponent

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Yongqiang Fu ◽  
Yingying Shan
Keyword(s):  
Singular Point ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Singular Points ◽  
Sufficient Condition ◽  
Variable Exponents ◽  
Isolated Singularities

AbstractIn this paper, we study the problem of removable isolated singularities for elliptic equations with variable exponents. We give a sufficient condition for removability of the isolated singular point for the equations in

Review Lecture Periods of integrals and topology of algebraic varieties

1984 ◽  
Vol 391 (1801) ◽  
pp. 231-254
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Topology ◽  
Major Theme ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Algebraic Functions ◽  
Several Variables ◽  
And Topology ◽  
Satisfactory Theory

A major theme of nineteenth century mathematics was the study of integrals of algebraic functions of one variable. This culminated in Riemann’s introduction of the surfaces that bear his name and analysis of periods of integrals on cycles on the surface. The creation of a correspondingly satisfactory theory for functions of several variables had to wait on the development of algebraic topology and its application by Lefschetz to algebraic varieties. These results were refined by Hodge’s theory of harmonic integrals. A closer analysis of Hodge structures by P. A. Griffiths and P. Deligne in recent years has led to unexpectedly strong restrictions on the topology of the variety and to a diversity of other applications. This advance is closely linked to the study of variation of integrals under deformations, particularly in the neighbourhood of a singular point.

scholarly journals Algebraic degeneracy theorem for holomorphic mappings into smooth projective algebraic varieties

1981 ◽  
Vol 84 ◽  
pp. 209-218
Author(s):  
Yoshihiro Aihara ◽  
Seiki Mori
Keyword(s):  
General Position ◽  
Holomorphic Mapping ◽  
Holomorphic Mappings ◽  
Holomorphic Curve ◽  
Algebraic Varieties ◽  
Picard Theorem ◽  
Picard’S Theorem ◽  
Picard's Theorem ◽  
Algebraic Degeneracy

The famous Picard theorem states that a holomorphic mapping f: C → P1(C) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at most n + 1 hyperplanes in Pn(C) in general position, thus extending Picard’s theorem (n = 1). Recently, Fujimoto [3], Green [4] and [5] obtained many Picard type theorems using Borel’s methods for holomorphic mappings.

HARTOGS-TYPE EXTENSION THEOREM AND SINGULAR SETS OF SEPARATELY HOLOMORPHIC MAPPINGS ON COMPACT SETS WITH VALUES IN A WEAKLY BRODY HYPERBOLIC COMPLEX SPACE

2001 ◽  
Vol 12 (07) ◽  
pp. 857-865
Author(s):  
DO DUC THAI ◽  
NGUYEN THI TUYET MAI
Keyword(s):  
Complex Space ◽  
Extension Theorem ◽  
Holomorphic Mappings ◽  
Compact Sets ◽  
Singular Sets ◽  
Hyperbolic Complex

We give a Hartogs-type extension theorem for separately holomorphic mappings on compact sets into a weakly Brody hyperbolic complex space. Moreover, a generalization of Saint Raymond–Siciak theorem of the singular sets of separately holomorphic mappings with values in a weakly Brody hyperbolic complex space is given.

Triple points of unknotting discs and the Arf invariant of knots

1994 ◽  
Vol 116 (1) ◽  
pp. 119-129 ◽  
Author(s):  
Thomas Fiedler
Keyword(s):  
Triple Point ◽  
Critical Points ◽  
Normal Bundle ◽  
Morse Function ◽  
Euler Number ◽  
Double Points ◽  
Arf Invariant ◽  
Triple Points ◽  
Coordinate Functions

AbstractAn unknotting disc is the ‘trace’ in ℝ4 of a homotopy of a diagram of a knot in ℝ3, which shrinks the diagram to a point. In this paper we study unknotting discs which have as singularities only ordinary triple points. It turns out that the Arf invariant of the knot is determined by the number of triple points in which all three branches of the disc intersect pairwise with the same index. We call such a triple point coherent. This interpretation of the Arf invariant has a surprising consequence:Let S ⊂ ℝ4 be a taut immersed sphere which has as singularities only ordinary triple points. Then the number of coherent triple points in S is even. For example, it is easy to show that there is a taut immersed sphere S with Euler number six of the normal bundle and which has exactly three ordinary double points and no other singularities. So, our result implies that the three double points of S can not be pushed together to create an ordinary triple point without the appearance of new singularities.Here ‘taut’ means that the restriction of one of the coordinate functions on S has exactly two (non-degenerate) critical points, i.e. is a perfect Morse function.

scholarly journals Intrinsic properties of surfaces with singularities

2015 ◽  
Vol 26 (04) ◽  
pp. 1540008 ◽  
Author(s):  
Masaru Hasegawa ◽  
Atsufumi Honda ◽  
Kosuke Naokawa ◽  
Kentaro Saji ◽  
Masaaki Umehara ◽  
...  
Keyword(s):  
The Other ◽  
Intrinsic Properties ◽  
Isolated Singularities ◽  
Wave Fronts ◽  
Singular Sets ◽  
The One

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss–Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.

Sign in / Sign up

Export Citation Format

Share Document