scholarly journals Lipschitz Normal Embedding Among Superisolated Singularities

2019 ◽  
Author(s):  
Filip Misev ◽  
Anne Pichon
Keyword(s):  
Tangent Cone ◽  
Infinite Family ◽  
Riemannian Metric ◽  
Analytic Space ◽  
Hypersurface Singularity ◽  
Trivial Case ◽  
Hermitian Metric ◽  
Surface Singularities ◽  
Inner Metric ◽  
Complex Analytic Space

Abstract Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if its projectivized tangent cone has only ordinary singularities. This provides an infinite family of LNE singularities, which is radically different from the class of minimal singularities.

scholarly journals Holomorphic automorphisms and cancellation theorems

1981 ◽  
Vol 81 ◽  
pp. 91-103 ◽  
Author(s):  
Toshio Urata
Keyword(s):  
Analytic Space ◽  
Biholomorphic Mapping ◽  
Direct Factor ◽  
Positive Dimension ◽  
Analytic Subvariety ◽  
Holomorphic Automorphisms ◽  
Complex Analytic Space

Let X be a complex analytic space of positive dimension and A a complex analytic subvariety of X. We call A a direct factor of X if there exist a complex analytic space B and a biholomorphic mapping f: A × B → X such that, for some b ∊ B, f(a, b) = a on A, and a complex analytic space X to be primary if X has no direct factor, not equal to X itself, of positive dimension.

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

Moderately Discontinuous Homotopy

2021 ◽  
Author(s):  
Javier Fernández de Bobadilla ◽  
Sonja Heinze ◽  
Maria Pe Pereira
Keyword(s):  
Tangent Cone ◽  
Homotopy Theory ◽  
Homology Theory ◽  
Comparison Theorems ◽  
Homotopy Groups ◽  
Homotopy Invariance ◽  
Normal Surface ◽  
Algebraic Invariant ◽  
Associated Spaces ◽  
Inner Metric

Abstract We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The $k$-th MD homotopy group is a group $MDH^b_{\bullet }$ for any $b\in [1,\infty ]$ together with homomorphisms $MD\pi ^b\to MD\pi ^{b^{\prime}}$ for any $b\geq b^{\prime}$. We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For $b=1$, it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty $, the $MD$-homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case.

scholarly journals The Af condition and relative conormal spaces for functions with nonvanishing derivative

2019 ◽  
Vol 30 (10) ◽  
pp. 1950050
Author(s):  
Terence Gaffney ◽  
Antoni Rangachev
Keyword(s):  
Analytic Function ◽  
Analytic Space ◽  
Numerical Criterion ◽  
Join Construction ◽  
Complex Analytic Space ◽  
Text Condition

We introduce a join construction, as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom’s [Formula: see text] condition.

scholarly journals Lowest weights in cohomology of variations of Hodge structure

2012 ◽  
Vol 206 ◽  
pp. 1-24
Author(s):  
Chris Peters ◽  
Morihiko Saito
Keyword(s):  
Open Subset ◽  
Hodge Structure ◽  
Analytic Space ◽  
Mixed Hodge Structure ◽  
Zariski Open Subset ◽  
Weight Part ◽  
Variation Of Hodge Structure ◽  
Local Monodromy ◽  
Variations Of Hodge Structure ◽  
Complex Analytic Space

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

ON THE DERIVATION LIE ALGEBRAS OF FEWNOMIAL SINGULARITIES

2018 ◽  
Vol 98 (1) ◽  
pp. 77-88 ◽  
Author(s):  
NAVEED HUSSAIN ◽  
STEPHEN S.-T. YAU ◽  
HUAIQING ZUO
Keyword(s):  
Lie Algebras ◽  
Hypersurface Singularity ◽  
Isolated Singularity ◽  
Solvable Algebra ◽  
Finite Dimensional ◽  
Hypersurface Singularities ◽  
Surface Singularities ◽  
Moduli Algebra ◽  
Upper Estimate ◽  
Appl Math

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra, $L(V)$, is the Lie algebra of derivations of the moduli algebra of $V$. It is a finite-dimensional solvable algebra and its dimension $\unicode[STIX]{x1D706}(V)$ is the Yau number. Fewnomial singularities are those which can be defined by an $n$-nomial in $n$ indeterminates. Yau and Zuo [‘A sharp upper estimate conjecture for the Yau number of weighted homogeneous isolated hypersurface singularity’, Pure Appl. Math. Q.12(1) (2016), 165–181] conjectured a bound for the Yau number and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper, we verify this conjecture for weighted homogeneous fewnomial surface singularities.

scholarly journals On Meromorphic Mappings into Taut Complex Analytic Spaces

1973 ◽  
Vol 50 ◽  
pp. 49-65
Author(s):  
Toshio Urata
Keyword(s):  
Holomorphic Mapping ◽  
Singular Points ◽  
Holomorphic Mappings ◽  
Analytic Space ◽  
Meromorphic Mapping ◽  
Analytic Set ◽  
Meromorphic Mappings ◽  
Proper Modification ◽  
Complex Analytic Space

In this paper, we study a certain difference between meromorphic mappings and holomorphic mappings into taut complex analytic spaces. We prove in §2 that, for any complex analytic space X, there exists a unique proper modification of X with center Sg (X) which is minimal with respect to the property that M(X) is normal and, for any T-meromorphic mapping f: X → Y (see Definition 1.3) into a complex analytic space Y, there exists a unique holomorphic mapping such that except some nowhere dense complex analytic set, where Sg(X) denotes the set of all singular points of X.

scholarly journals \'Etale triviality of finite equivariant vector bundles

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Indranil Biswas ◽  
Peter O'Sullivan
Keyword(s):  
Vector Bundle ◽  
Lie Group ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Regular Function ◽  
Analytic Space ◽  
Complex Analytic Space

Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{\mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite \'etale covering of X is trivial as an H-equivariant bundle.

Differential Forms and Resolutions on Certain Analytic Spaces II. Flat Resolutions

1992 ◽  
Vol 44 (4) ◽  
pp. 728-749 ◽  
Author(s):  
Vincenzo Ancona ◽  
Bernard Gaveau
Keyword(s):  
General Result ◽  
Exceptional Divisor ◽  
Differential Forms ◽  
Singular Locus ◽  
Analytic Space ◽  
The One ◽  
Complex Analytic Space

This paper gives another construction of (0, p)-forms on a complex analytic space and of the operator. This construction is independent of the one in [1] and apart from the general result of Section 1 of [1], it can be read independently. As in [1], the hypotheses on S are the following: S has normal singularities, its singular locus X is smooth, the exceptional divisor in a desingularization of S is irreducible.

Sign in / Sign up

Export Citation Format

Share Document