scholarly journals Holomorphic and differentiable tangent spaces to a complex analytic variety

1977 ◽  
Vol 12 (3) ◽  
pp. 377-401
Author(s):  
Joseph Becker
Keyword(s):  
Analytic Variety ◽  
Complex Analytic Variety

scholarly journals Goresky–Pardon lifts of Chern classes and associated Tate extensions

2017 ◽  
Vol 153 (7) ◽  
pp. 1349-1371 ◽  
Author(s):  
Eduard Looijenga
Keyword(s):  
Vector Bundle ◽  
Hodge Structure ◽  
Holomorphic Vector Bundle ◽  
Mixed Hodge Structure ◽  
Chern Classes ◽  
Analytic Variety ◽  
Algebraic Setting ◽  
Geometric Conditions ◽  
Stable Cohomology ◽  
Complex 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.

Green's Forms and Meromorphic Functions on Compact Analytic Varieties

1951 ◽  
Vol 3 ◽  
pp. 108-128 ◽  
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).

scholarly journals Mixed Bruce–Roberts numbers

2020 ◽  
Vol 63 (2) ◽  
pp. 456-474 ◽  
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.

Local fiberings of a complex analytic variety

1967 ◽  
Vol 172 (4) ◽  
pp. 313-326
Author(s):  
Thomas Bloom
Keyword(s):  
Analytic Variety ◽  
Complex Analytic Variety

NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

2017 ◽  
Vol 60 (1) ◽  
pp. 175-185 ◽  
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.

A FAMILY OF SURFACES CONSTRUCTED FROM GENUS 2 CURVES

2007 ◽  
Vol 18 (05) ◽  
pp. 585-612 ◽  
Author(s):  
CHAD SCHOEN
Keyword(s):  
Riemann Surface ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
Analytic Variety ◽  
Genus 2 ◽  
Complex Analytic Variety ◽  
Projective Manifolds ◽  
Genus 2 Curves ◽  
Complex Projective

We consider the deformations of the two-dimensional complex analytic variety constructed from a genus 2 Riemann surface by attaching its self-product to its Jacobian in an elementary way. The deformations are shown to be unobstructed, the variety smooths to give complex projective manifolds whose invariants are computed and whose images under Albanese maps (re)verify an instance of the Hodge conjecture for certain abelian fourfolds.

scholarly journals HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES

2004 ◽  
Vol 15 (09) ◽  
pp. 895-905 ◽  
Author(s):  
W. EBELING ◽  
S. M. GUSEIN-ZADE ◽  
J. SEADE
Keyword(s):  
Complete Intersection ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Analytic Variety ◽  
Curve Singularities ◽  
Complex Analytic Variety ◽  
Arbitrary Curve

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

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