The Gauss–Bonnet defect of complex affine hypersurfaces

Siersma, Dirk; Tibăr, Mihai

doi:10.1016/j.bulsci.2005.08.003

Cited by 8 publications

(4 citation statements)

References 16 publications

(18 reference statements)

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“…However, using arguments from differential topology and integral geometry, one sees that these invariants admit geometric characterizations that still make sense in the real case. For instance, the multiplicity of a complex analytic germ is equal to its density [Dra] and the μ * -sequence, the polar multiplicities and the generic polar intersection multiplicities are related to curvature integrals (see [La,Loe,Dut1,SiTi2]). Unfortunately, in the real situation, these geometric quantities do not belong to discrete sets and therefore, one cannot expect results relating their constancy to regularity conditions.…”

Section: Introductionmentioning

confidence: 99%

Gauss–Kronecker curvature and equisingularity at infinity of definable families

Dutertre

Grandjean

2021

Asian Journal of Mathematics

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Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let (Ts) s∈R be a definable family of C 2 -hypersurfaces of R n . Upon defining the notion of generalized critical value for such a family, we show that the functions s → |K|(s) and s → K(s), respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of Ts, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.

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Section: Introductionmentioning

confidence: 99%

Gauss–Kronecker curvature and equisingularity at infinity of definable families

Dutertre

Grandjean

2021

Asian Journal of Mathematics

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show abstract

“…However, using arguments from differential topology and integral geometry, one sees that these invariants admit geometric characterizations that still make sense in the real case. For instance, the multiplicity of a complex analytic germ is equal to its density [Dra] and the µ * -sequence, the polar multiplicities and the generic polar intersection multiplicities are related to curvature integrals (see [La,Loe,Dut1,SiTi2]). Unfortunately, in the real situation, these geometric quantities do not belong to discrete sets and therefore, one cannot expect results relating their constancy to regularity conditions.…”

Section: Introductionmentioning

confidence: 99%

Gauss-Kronecker Curvature and equisingularity at infinity of definable families

Dutertre,

Grandjean

2019

Preprint

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Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let (Ts) s∈R be a globally definable family of C 2 -hypersurfaces of R n . Upon defining the notion of generalized critical value for such a family, we show that the functions s → |K(s)| and s → K(s), respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of Ts, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.

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“…Our Theorems C and D are, in a way, a continuation of Langevin [10] (total curvature), Garcia Barosso and Teissier [1] (concentration of curvature). The work of Siersma and Tibar [14] is in a different direction.…”

Section: Introductionmentioning

confidence: 99%

A’Campo curvature bumps and the Dirac phenomenon near a singular point

Koike¹,

Kuo²,

Păunescu³

2015

Proc. London Math. Soc.

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The level curves of an analytic function germ can have bumps (maxima of Gaussian curvature) at unexpected points near the singularity. This phenomenon is fully explored for f(z,w)∈C{z,w}, using the Newton–Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac phenomenon: as c→0, the total Gaussian curvature of f=c accumulates in the minimal gradient canyons, and nowhere else. Our approach mimics the introduction of polar coordinates in Analytic Geometry.

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The Gauss–Bonnet defect of complex affine hypersurfaces

Abstract: We study the loss of curvature at the "ends" of a hypersurface in the affine space and we express it in terms of singularities at infinity.

Cited by 8 publications

References 16 publications

Gauss–Kronecker curvature and equisingularity at infinity of definable families

Gauss–Kronecker curvature and equisingularity at infinity of definable families

Gauss-Kronecker Curvature and equisingularity at infinity of definable families

A’Campo curvature bumps and the Dirac phenomenon near a singular point

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