Cyclic covers of Pv branched along v + 2 hyperplanes and the generalized Hodge Conjecture for certain abelian varieties

1989 ◽  
pp. 137-154 ◽  
Author(s):  
Chad Schoen
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture ◽  
Cyclic Covers ◽  
Generalized Hodge Conjecture

scholarly journals Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes

2014 ◽  
Vol 66 (3) ◽  
pp. 505-524 ◽  
Author(s):  
Donu Arapura
Keyword(s):  
Projective Space ◽  
Prime Number ◽  
Hyperplane Arrangement ◽  
Hodge Theory ◽  
Cyclic Cover ◽  
Hodge Conjecture ◽  
Cyclic Covers ◽  
Generalized Hodge Conjecture ◽  
Ramification Locus

AbstractSuppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D and that U is the complement of the ramification locus in Y. The first theorem in this paper implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance, this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.

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.

ABELIAN VARIETIES OF TYPE III AND THE HODGE CONJECTURE

1999 ◽  
Vol 10 (06) ◽  
pp. 667-675 ◽  
Author(s):  
SALMAN ABDULALI
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture ◽  
Type Iii ◽  
General Member

We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.

scholarly journals Notes on Absolute Hodge Classes

2017 ◽  
Author(s):  
Franc¸ois Charles ◽  
Christian Schnell
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture ◽  
De Rham Cohomology ◽  
Galois Action ◽  
De Rham ◽  
Definition Of ◽  
Full Proof

This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.

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