• Corpus ID: 235458442

On the Transcendence of Period Images

@inproceedings{Urbanik2021OnTT,
  title={On the Transcendence of Period Images},
  author={David Urbanik},
  year={2021}
}
Let f : X Ñ S be a family of smooth projective algebraic varieties over a smooth connected base S, with everything defined over Q. Denote by V “ Rf ̊Zpiq the associated integral variation of Hodge structure on the degree 2i cohomology. We consider the following question: when can a fibre Vs above an algebraic point s P SpQq be isomorphic to a transcendental fibre Vs1 with s 1 P SpCqzSpQq? When V induces a quasi-finite period map φ : S Ñ ΓzD, conjectures in Hodge theory predict that such… 
2 Citations
Sets of Special Subvarieties of Bounded Degree
Let f : X Ñ S be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base S, and let V “ R2kf ̊Zpkq be the integral variation of Hodge structure coming from
Effective Methods for Diophantine Finiteness
Let K Ă C be a number field, and let OK,N “ OKrN s be its ring of N-integers. Recently, Lawrence and Venkatesh proposed a general strategy for proving the Shafarevich conjecture for the fibres of a

References

SHOWING 1-10 OF 39 REFERENCES
Tame topology of arithmetic quotients and algebraicity of Hodge loci
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex
Rigidity of Spreadings and Fields of Definition
Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely˘i curves and Shimura varieties. Rigidity is related to
NÉRON MODELS
§1.1. Motivation. The purpose of these notes is to explain the definition and basic properties of the Néron model A of an abelian variety A over a global or local field K. We also give some idea of
Lie groups and Lie algebras
The first example of a Lie group is the general linear group GL(n,R) = {A ∈ Matn(R)| det(A) 6= 0} of invertible n × n matrices. It is an open subset of Matn(R), hence a submanifold, and the
Local Theory of Complex Spaces
TLDR
This introductory chapter is a rambling through basic notions and results of Local Complex Analysis based on local function theory, local algebra and sheaves.
The Ax–Schanuel conjecture for variations of Hodge structures
We extend the Ax–Schanuel theorem recently proven for Shimura varieties by Mok–Pila–Tsimerman to all varieties supporting a pure polarizable integral variation of Hodge structures. In fact, Hodge
Jet schemes and singularities
We give a self-contained presentation of the basic results on jet schemes of singular varieties. Applications are given to invariants of singularities, such as minimal log discrepancies. We simplify
On the closure of the Hodge locus of positive period dimension
Given $${{\mathbb {V}}}$$ a polarizable variation of $${{\mathbb {Z}}}$$ -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$
Techniques de construction en géométrie analytique. VI. Étude locale des morphismes : germes d'espaces analytiques, platitude, morphismes simples
© Séminaire Henri Cartan (Secrétariat mathématique, Paris), 1960-1961, tous droits réservés. L’accès aux archives de la collection « Séminaire Henri Cartan » implique l’accord avec les conditions
...
1
2
3
4
...