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2015 | OriginalPaper | Chapter

Algebraic and Arithmetic Properties of Period Maps

Author: Matt Kerr

Published in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Publisher: Springer New York

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Abstract

We survey recent developments in Hodge theory which are closely tied to families of CY varieties, including Mumford-Tate groups and boundary components, as well as limits of normal functions and generalized Abel-Jacobi maps. While many of the techniques are representation-theoretic rather than motivic, emphasis is placed throughout on the (known and conjectural) arithmetic properties accruing to geometric variations.
Footnotes
1
That is, its finite-dimensional representations are reducible.
 
2
Of course, one can play the same game with \(\underline{h} = (1,n,n, 1)\) and \(\underline{h}_{+} = (1,n, 0, 0)\) to embed \(\mathbb{B}_{n}\) in a non-Hermitian period domain.
 
3
Here \(\mathbb{V}\) is a \(\mathbb{Q}\)-local system, \(\mathcal{V}:= \mathbb{V} \otimes \mathcal{O}_{\mathcal{S}}\) the [sheaf of sections of the] holomorphic vector bundle, and \(\mathcal{F}^{\bullet }\) a filtration by [sheaves of sections of] holomorphic subbundles.
 
4
We thank C. Robles for providing this example.
 
5
That is, p is a point of maximal transcendence degree; equivalently, it lies in the complement of the complex points of countably many \(\bar{\mathbb{Q}}\)-subvarieties.
 
6
Added in proof: By a recent result of M. Yoshinaga (arXiv:0805.0349v1), it turns out that the non-elementary real numbers cannot be real or imaginary parts of periods in the sense of Kontsevich and Zagier. For the period domain D = D (1, 0, 0, 1), the spread argument shows that the period ratio of any motivic HS in D is a K-Z period, solving the problem as stated (take τ to be \(\sqrt{-1}\) times a non-elementary real number). So the problem should be reformulated to ask whether one has elementary non-motivic Hodge structures, i.e. ones all of whose periods have elementary real and imaginary parts. (We thank W. Xiuli for pointing out Yoshinaga’s article.)
 
7
While there is nothing conjectural about our construction of F BB , the existence of a “Bloch-Beilinson filtration” is conjectural. Our F BB only qualifies as one if \(\cap _{i}F_{BB}^{i} =\{ 0\}\); this depends on the injectivity of Ψ, which is sometimes called the “arithmetic Bloch-Beilinson conjecture”.
 
8
This is just (an arithmetic quotient of) a M-T domain for HS of level one cut out by 2-tensors.
 
9
Note: dropping the \(\mathbb{Z}\) will mean \(\mathbb{Q}\)-coefficients.
 
10
This can be done over \(\mathbb{R}\) precisely when the LMHS is \(\mathbb{R}\)-split, i.e. \(I^{p,q} = \overline{I^{q,p}}\) exactly.
 
11
This class can be derived from work of Iritani [20, 31].
 
12
Their theorem applies to the more general setting \([\mathfrak{z}\vert _{\mathcal{X}^{{\ast}}}] = 0\) in \(H^{2m}(\mathcal{X}^{{\ast}})\).
 
13
There is a related important construction of Schnell which leads to a very natural proof of the algebraicity of 0-loci of normal functions [51]. Also note that, while Hausdorff, \(\hat{\mathcal{J}}_{e}\) may not be a complex analytic space: the fiber over 0 usually has lower dimension than the other fibers (cf. Sect. 4.2.5).
 
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Metadata
Title
Algebraic and Arithmetic Properties of Period Maps
Author
Matt Kerr
Copyright Year
2015
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-2830-9_6

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