scholarly journals SINGULARITIES OF THE BIEXTENSION METRIC FOR FAMILIES OF ABELIAN VARIETIES

2018 ◽  
Vol 6 ◽  
Author(s):  
JOSÉ IGNACIO BURGOS GIL ◽  
DAVID HOLMES ◽  
ROBIN DE JONG
Keyword(s):  
Abelian Varieties ◽  
Arbitrary Dimension ◽  
Invariant Metric ◽  
Hodge Structures

In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$.

scholarly journals Hodge Structures on Abelian Varieties of Type III

2002 ◽  
Vol 155 (3) ◽  
pp. 915 ◽  
Author(s):  
Salman Abdulali
Keyword(s):  
Abelian Varieties ◽  
Hodge Structures ◽  
Type Iii

scholarly journals THE KUGA–SATAKE CONSTRUCTION UNDER DEGENERATION

2019 ◽  
Vol 19 (6) ◽  
pp. 2165-2182
Author(s):  
Stefan Schreieder ◽  
Andrey Soldatenkov
Keyword(s):  
Abelian Varieties ◽  
Hodge Theory ◽  
Hodge Structures ◽  
Mixed Hodge Structures

We extend the Kuga–Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga–Satake abelian varieties associated with polarized variations of K3 type Hodge structures over the punctured disc.

Hodge structures on abelian varieties of type IV

2004 ◽  
Vol 246 (1-2) ◽  
pp. 203-212 ◽  
Author(s):  
Salman Abdulali
Keyword(s):  
Abelian Varieties ◽  
Hodge Structures ◽  
Type Iv

scholarly journals On the closure of the Hodge locus of positive period dimension

2021 ◽  
Author(s):  
B. Klingler ◽  
A. Otwinowska
Keyword(s):  
Moduli Space ◽  
Classical Result ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Countable Union ◽  
Adjoint Group ◽  
Hodge Structures ◽  
Period Map ◽  
Tate Group ◽  
Irreducible Components

AbstractGiven $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

Subgroups and subrings of profinite rings

1994 ◽  
Vol 116 (2) ◽  
pp. 209-222 ◽  
Author(s):  
A. G. Abercrombie
Keyword(s):  
Topological Group ◽  
Hausdorff Dimension ◽  
Haar Measure ◽  
Arbitrary Dimension ◽  
Invariant Probability Measure ◽  
Fractional Dimension ◽  
Invariant Metric ◽  
Analogous Statement ◽  
Natural Choice ◽  
Carry Over

AbstractA profinite topological group is compact and therefore possesses a unique invariant probability measure (Haar measure). We shall see that it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric. This fractional dimension is analogous to Hausdorff dimension in ℝ.It is therefore natural to ask to what extent known results concerning Hausdorff dimension in ℝ carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in [12], we consider rings of a-adic integers and investigate the possible dimensions of their subgroups and subrings. We will find that for each prime p the ring of p-adic integers possesses subgroups of arbitrary dimension. This should cause little surprise since a similar result is known to hold in ℝ. However, we will also find that there exists a ring of a-adic integers possessing Borel subrings of arbitrary dimension. This is in contrast with the situation in ℝ, where the analogous statement is known to be false.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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