On the singularity of an analytic variety

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

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Green's Forms and Meromorphic Functions on Compact Analytic Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-1951-014-1 ◽

1951 ◽

Vol 3 ◽

pp. 108-128 ◽

Cited By ~ 2

Author(s):

Kunihiko Kodaira

Keyword(s):

Meromorphic Function ◽

Meromorphic Functions ◽

Zero Point ◽

Positive Definite ◽

Analytic Surface ◽

Analytic Variety ◽

Complex Dimension ◽

Analytic Varieties ◽

Complex Analytic Variety ◽

Zero Points

Let be a compact complex analytic variety of the complex dimension n with a positive definite Kâhlerian metric [4] ; the local analytic coordinates on will be denoted by z = (z 1 z 2, … , zn). Now, suppose a meromorphic function f(z) defined on as given. Then the poles and zero-points of f(z) constitute an analytic surface in consisting of a finite number of irreducible closed analytic surfaces Γ1, Γ2, … , Γk, each of which is a polar or a zero-point variety of f(z).

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Mixed Bruce–Roberts numbers

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091519000543 ◽

2020 ◽

Vol 63 (2) ◽

pp. 456-474 ◽

Cited By ~ 1

Author(s):

Carles Bivià-Ausina ◽

Maria Aparecida Soares Ruas

Keyword(s):

Analytic Function ◽

Function Germ ◽

Analytic Variety ◽

Fundamental Properties ◽

Complex Analytic Variety

AbstractWe extend the notions of μ*-sequences and Tjurina numbers of functions to the framework of Bruce–Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X ⊆ ℂn and a finitely ${\mathcal R}(X)$-determined analytic function germ f : (ℂn, 0) → (ℂ, 0). We analyze some fundamental properties of these numbers.

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Local fiberings of a complex analytic variety

Mathematische Annalen ◽

10.1007/bf01351594 ◽

1967 ◽

Vol 172 (4) ◽

pp. 313-326

Author(s):

Thomas Bloom

Keyword(s):

Analytic Variety ◽

Complex Analytic Variety

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analytic variety in hyperspace

Duke Mathematical Journal ◽

10.1215/s0012-7094-48-01523-3 ◽

1948 ◽

Vol 15 (1) ◽

pp. 207-218

Author(s):

P. O. Bell

Keyword(s):

Analytic Variety

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Tangents to an Analytic Variety

Annals of Mathematics ◽

10.2307/1970400 ◽

1965 ◽

Vol 81 (3) ◽

pp. 496 ◽

Cited By ~ 173

Author(s):

Hassler Whitney

Keyword(s):

Analytic Variety

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NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

Glasgow Mathematical Journal ◽

10.1017/s0017089516000641 ◽

2017 ◽

Vol 60 (1) ◽

pp. 175-185 ◽

Cited By ~ 3

Author(s):

J. J. NUÑO-BALLESTEROS ◽

B. ORÉFICE-OKAMOTO ◽

J. N. TOMAZELLA

Keyword(s):

Sufficient Conditions ◽

Additional Term ◽

Milnor Number ◽

Curve Singularity ◽

Hypersurface Singularity ◽

Weighted Degree ◽

Function Germ ◽

Analytic Variety ◽

Complex Analytic Variety ◽

Necessary And Sufficient

AbstractWe consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

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