The étale site of a rigid analytic variety and an adic space

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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