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Algebraic and Arithmetic Properties of Period Maps

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Book cover Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Part of the book series: Fields Institute Monographs ((FIM,volume 34))

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.

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Notes

  1. 1.

    That is, its finite-dimensional representations are reducible.

  2. 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. 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. 4.

    We thank C. Robles for providing this example.

  5. 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. 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. 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. 8.

    This is just (an arithmetic quotient of) a M-T domain for HS of level one cut out by 2-tensors.

  9. 9.

    Note: dropping the \(\mathbb{Z}\) will mean \(\mathbb{Q}\)-coefficients.

  10. 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. 11.

    This class can be derived from work of Iritani [20, 31].

  12. 12.

    Their theorem applies to the more general setting \([\mathfrak{z}\vert _{\mathcal{X}^{{\ast}}}] = 0\) in \(H^{2m}(\mathcal{X}^{{\ast}})\).

  13. 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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Acknowledgements

The author acknowledges partial support under the aegis of NSF Grant DMS-1068974. It is his pleasure to thank C. Robles and the referee for their constructive comments on the manuscript.

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  1. Department of Mathematics, Washington University in St. Louis, Campus Box 1146, St. Louis, MO, 63130, USA

    Matt Kerr

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  1. Mathematics Department, Stony Brook University, Stony Brook, New York, USA

    Radu Laza

  2. Leibniz Universität Hannover, Hannover, Germany

    Matthias Schütt

  3. Department of Math & Stats, Queen's University, Kingston, Ontario, Canada

    Noriko Yui

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Kerr, M. (2015). Algebraic and Arithmetic Properties of Period Maps. In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2830-9_6

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