scholarly journals On the fundamental groups of normal varieties

2016 ◽  
Vol 18 (04) ◽  
pp. 1550065 ◽  
Author(s):  
Donu Arapura ◽  
Alexandru Dimca ◽  
Richard Hain
Keyword(s):  
Fundamental Groups ◽  
Algebraic Varieties ◽  
Local Systems ◽  
Normal Variety ◽  
Normal Complex ◽  
Normal Varieties ◽  
Rank One

We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.

scholarly journals A Note on the Differential Forms on Everywhere Normal Varieties

1951 ◽  
Vol 2 ◽  
pp. 93-94
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Algebraic Geometry ◽  
Differential Form ◽  
Differential Forms ◽  
Multiple Point ◽  
Algebraic Varieties ◽  
Simple Point ◽  
Multiple Points ◽  
Complete Variety ◽  
Normal Variety ◽  
Normal Varieties

A. Weil proposed in his book “Foundations of algebraic geometry” several problems concerning differential forms on algebraic varieties, S. Koizumi has proved that if ω is a differential form on a complete variety U without multiple point, which is finite at every point of IT, then ω is the differential form of the first kind. The following example shows that on everywhere normal varieties with multiple points this statement holds no more; that is: A differential form on a everywhere normal variety which is finite on every simple point of its variety is not always the differential form of the first kind.

scholarly journals Topology of Subvarieties of Complex Semi-abelian Varieties

2020 ◽  
Author(s):  
Yongqiang Liu ◽  
Laurentiu Maxim ◽  
Botong Wang
Keyword(s):  
Euler Characteristic ◽  
Morse Theory ◽  
Characteristic Property ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Local Systems ◽  
Affine Manifolds ◽  
Cyclic Covers ◽  
Rank One ◽  
Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

scholarly journals GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

2013 ◽  
Vol 1 ◽  
Author(s):  
MIHNEA POPA ◽  
CHRISTIAN SCHNELL
Keyword(s):  
Abelian Varieties ◽  
Local Systems ◽  
Coherent Sheaves ◽  
Natural Categories ◽  
Rank One ◽  
Generic Vanishing ◽  
Mixed Hodge Modules ◽  
Irregular Varieties

AbstractWe extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

scholarly journals Fundamental groups of open algebraic varieties

Topology ◽  
1995 ◽  
Vol 34 (3) ◽  
pp. 509-531 ◽  
Author(s):  
Ichiro Shimada
Keyword(s):  
Fundamental Groups ◽  
Algebraic Varieties

A GENERALIZATION OF LEFSCHETZ-ZARISKI THEOREM ON FUNDAMENTAL GROUPS OF ALGEBRAIC VARIETIES

1995 ◽  
Vol 06 (06) ◽  
pp. 921-932 ◽  
Author(s):  
ICHIRO SHIMADA
Keyword(s):  
Fundamental Groups ◽  
Natural Homomorphism ◽  
Algebraic Varieties

Let X and Y be the complements of divisors on non-singular irreducible closed subvarieties [Formula: see text] and [Formula: see text] in ℙn, respectively. Suppose that dim X+ dim Y≥n+2. Then, for a general g∈PGL (n+1), the natural homomorphism π1(g(X)∩Y)→π1(Y) induces a surjection from Ker (π1(g(X)∩Y)→π1(g(X))) onto π1(Y), and there is a surjection to its kernel from the cokernel of π2(X)→π2(ℙn). In particular, if E⊂ℙn is a hypersurface and 2·dim [Formula: see text], then [Formula: see text] is isomorphic to π1(ℙn\E) for a general g∈PGL (n+1).

scholarly journals Étale cohomology of rank one $$\ell $$-adic local systems in positive characteristic

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Hélène Esnault ◽  
Moritz Kerz
Keyword(s):  
Positive Characteristic ◽  
Vanishing Theorem ◽  
Deformation Spaces ◽  
Local Systems ◽  
Etale Cohomology ◽  
Lefschetz Theorem ◽  
Rank One ◽  
Étale Cohomology ◽  
Generic Vanishing ◽  
Hard Lefschetz Theorem

AbstractWe show that in positive characteristic special loci of deformation spaces of rank one $$\ell $$ ℓ -adic local systems are quasi-linear. From this we deduce the Hard Lefschetz theorem for rank one $$\ell $$ ℓ -adic local systems and a generic vanishing theorem.

scholarly journals On the Theory of Differential Forms on Algebraic Varieties

1957 ◽  
Vol 11 ◽  
pp. 13-39
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Function Field ◽  
Projective Variety ◽  
Hyperplane Section ◽  
Differential Forms ◽  
Classical Case ◽  
Algebraic Varieties ◽  
Simple Point ◽  
Perfect Field ◽  
Linearly Independent ◽  
Normal Varieties

Let K be a function field of one variable over a perfect field k and let v be a valuation of K over k. Then is the different-divisor (Verzweigungsdivisor) of K/k(x), and (x)∞ is the denominator-divisor (Nennerdivisor) of x. In §1 we consider a generalization of this theorem in the function fields of many variables under some conditions. In §2 and §3 we consider the differential forms of the first kind on algebraic varieties, or the differential forms which are finite at every simple point of normal varieties and subadjoint hypersurfaces which are developed by Clebsch and Picard in the classical case. In §4 we give a proof of the following theorem. Let Vr be a normal projective variety defined over a field k of characteristic 0, and let ω1, …, ωs be linearly independent simple closed differential forms which are finite at every simple point of Vr. Then the induced forms on a generic hyperplane section are also linearly independent.

scholarly journals Rank One Local Systems and Forms of Degree One

2015 ◽  
Vol 2016 (13) ◽  
pp. 3849-3855 ◽  
Author(s):  
Nero Budur ◽  
Botong Wang ◽  
Youngho Yoon
Keyword(s):  
Local Systems ◽  
Rank One
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