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Published in:

2015 | OriginalPaper | Chapter

# Algebraic and Arithmetic Properties of Period Maps

Author: Matt Kerr

Publisher: Springer New York

## 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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