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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
- 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).
References
Abdulali, S.: Hodge structures of CM type. J. Ramanujan Math. Soc. 20(2), 155–162 (2005)
Allcock, D., Carlson, J., Toledo, D.: The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebr. Geom. 11(4), 659–724 (2002)
Allcock, D., Carlson, J., Toledo, D.: The moduli space of cubic threefolds as a ball quotient. Mem. Am. Math. Soc. 209(985), xii+70 (2011)
André, Y.: Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82(1), 1–24 (1992)
André, Y.: Galois theory, motives, and transcendental numbers (2008, preprint). arXiv:0805.2569
Arapura, D., Kumar, M.: Beilinson-Hodge cycles on semiabelian varieties. Math. Res. Lett. 16(4), 557–562 (2009)
Brosnan, P., Pearlstein, G.: On the algebraicity of the zero locus of an admissible normal function. Compositio Math. 149, 1913–1962 (2013)
Brosnan, P., Fang, H., Nie, Z., Pearlstein, G.: Singularities of normal functions. Invent. Math. 177, 599–629 (2009)
Candelas, P., de la Ossa, X., Green, P., Parkes, L.: A pair of manifolds as an exactly solvable superconformal theory. Nucl. Phys. B359, 21–74 (1991)
Carlson, J.: Bounds on the dimensions of variations of Hodge structure. Trans. Am. Math. Soc. 294, 45–64 (1986); Erratum, Trans. Am. Math. Soc. 299, 429 (1987)
Cattani, E.: Mixed Hodge Structures, Compactifications and Monodromy Weight Filtration. Annals of Math Study, vol. 106, pp. 75–100. Princeton University Press, Princeton (1984)
Cattani, E., Deligne, P., Kaplan, A.: On the locus of Hodge classes. JAMS 8, 483–506 (1995)
Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. 123, 457–535 (1986)
Charles, F.: On the zero locus of normal functions and the étale Abel-Jacobi map. IMRN 2010(12), 2283–2304 (2010)
Clingher, A., Doran, C.: Modular invariants for lattice polarized K3 surfaces. Mich. Math. J. 55(2), 355–393 (2007)
Cohen, P.: Humbert surfaces and transcendence properties of automorphic functions. Rocky Mt. J. Math. 26(3), 987–1001 (1996)
da Silva, G., Jr., Kerr, M., Pearlstein, G.: Arithmetic of degenerating principal VHS: examples arising from mirror symmetry and middle convolution. Can. J. Math (2015 to appear)
Deligne, P.: Hodge cycles on abelian varieties (Notes by J. S. Milne). In: Deligne, P. (ed.) Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900, pp. 9–100. Springer, New York (1982)
Dettweiler, M., Reiter, S.: Rigid local systems and motives of type G 2. Compositio Math. 146, 929–963 (2010)
Doran, C., Kerr, M.: Algebraic cycles and local quantum cohomology Commun. Number Theory Phys. 8, 703–727 (2014)
Doran, C., Morgan, J.: Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi-Yau threefolds. In: Mirror Symmetry V: Proceedings of the BIRS Workshop on CY Varieties and Mirror Symmetry, Banff, Dec 2003. AMS/IP Studies in Advanced Mathematics, vol. 38, pp. 517–537. AMS, Providence (2006)
El Zein, F., Zucker, S.: Extendability of normal functions associated to algebraic cycles. In: Griffiths, P. (ed.) Topics in Transcendental Algebraic Geometry. Annals of Mathematics Studies, vol. 106. Princeton University Press, Princeton (1984)
Friedman, R., Laza, R.: Semi-algebraic horizontal subvarieties of Calabi-Yau type (2011, preprint). To appear in Duke Math. J., available at arXiv:1109.5632
Green, M., Griffiths, P.: Algebraic cycles and singularities of normal functions. In: Nagel, J., Peters, C. (eds.) Algebraic Cycles and Motives. LMS Lecture Note Series, vol. 343, pp. 206–263, Cambridge University Press, Cambridge (2007)
Green, M., Griffiths, P., Kerr, M.: Neron models and boundary components for degenerations of Hodge structures of mirror quintic type. In: Alexeev, V. (ed.) Curves and Abelian Varieties. Contemporary Mathematics, vol. 465, pp. 71–145. AMS, Providence (2007)
Green, M., Griffiths, P., Kerr, M.: Néron models and limits of Abel-Jacobi mappings. Compositio Math. 146, 288–366 (2010)
Green, M., Griffiths, P., Kerr, M.: Mumford-Tate Groups and Domains. Their Geometry and Arithmetic. Annals of Mathematics Studies, vol. 183. Princeton University Press, Princeton (2012)
Griffiths, P., Robles, C., Toledo, D.: Quotients of non-classical flag domains are not algebraic (2013, preprint). Available at arXiv:1303.0252
Hazama, F.: Hodge cycles on certain abelian varieties and powers of special surfaces. J. Fac. Sci. Univ. Tokyo Sect. 1a 31, 487–520 (1984)
Holzapfel, R.-P.: The Ball and Some Hilbert Problems. Birkhäuser, Basel (1995)
Iritani, H.: An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Adv. Math. 222(3), 1016–1079 (2009)
Jefferson, R., Walcher, J.: Monodromy of inhomogeneous Picard-Fuchs equations (2013, preprint). arXiv:1309.0490
Kato, K., Usui, S.: Classifying Spaces of Degenerating Polarized Hodge Structure. Annals of Mathematics Studies, vol. 169, Princeton University Press, Princeton (2009)
Katz, N.: Rigid Local Systems. Princeton University Press, Princeton (1995)
Kerr, M., Lewis, J.: The Abel-Jacobi map for higher Chow groups, II. Invent. Math. 170, 355–420 (2007)
Kerr, M., Pearlstein, G.: An exponential history of functions with logarithmic growth. In: Friedman, G. et al. (eds.) Topology of Stratified Spaces. MSRI Publications, vol. 58. Cambridge University Press, New York (2011)
Kerr, M., Pearlstein, G.: Boundary Components of Mumford-Tate Domains. Duke Math. J. (to appear). arXiv:1210.5301
Kerr, M., Pearlstein, G.: Naive boundary strata and nilpotent orbits. Ann. Inst. Fourier 64(6), 2659–2714 (2014)
Kondo, S.: The moduli space of curves of genus 4 and Deligne-Mostow’s complex reflection groups. Algebr. Geom. (2000). Azumino (Hotaka). Adv. Stud. Pure Math. 36, 383–400 (2002). The Mathematical Society of Japan, Tokyo
Laporte, G., Walcher, J.: Monodromy of an inhomogeneous Picard-Fuchs equation. SIGMA 8, 056, 10 (2012)
Lefschetz, S.: l’Analysis situs et la géometrié algébrique. Gauthier-Villars, Paris (1924)
Movasati, H.: Modular-type functions attached to mirror quintic Calabi-Yau varieties (2012, preprint). arXiv:1111.0357
Murty, V.K.: Exceptional Hodge classes on certain abelian varieties. Math. Ann. 268, 197–206 (1984)
Patrikis, S.: Mumford-Tate groups of polarizable Hodge structures (2013, preprint). arXiv:1302.1803
Peters, C., Steenbrink, J.: Mixed Hodge Structures. Ergebnisse der Mathematik Ser. 3, vol. 52. Springer, Berlin (2008)
Robles, C.: Schubert varieties as variations of Hodge structure (2012, preprint). arXiv:1208.5453, to appear in Selecta Math
Saito, M.: Hausdorff property of the Zucker extension at the monodromy invariant subspace (2008, preprint). arXiv:0803.2771
Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)
Schneider, T.: Einführung in die transzendenten Zahlen (German). Springer, Berlin (1957)
Schnell, C.: Two observations about normal functions. Clay Math. Proc. 9, 75–79 (2010)
Schnell, C.: Complex-analytic Néron models for arbitrary families of intermediate Jacobians. Invent. Math. 188(1), 1–81 (2012)
Shiga, H.: On the representation of the Picard modular function by θ constants I-II. Publ. RIMS Kyoto Univ. 24, 311–360 (1988)
Shiga, H., Wolfart, J.: Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. 463, 1–25 (1995)
Sommese, A.: Criteria for quasi-projectivity. Math. Ann. 217, 247–256 (1975)
Usui, S.: Generic Torelli theorem for quintic-mirror family. Proc. Jpn. Acad. Ser. A Math. Sci. 84(8), 143–146 (2008)
Voisin, C.: Hodge loci and absolute Hodge classes. Compos. Math. 143(4), 945–958 (2007)
Wüstholz, G.: Algebraic groups, Hodge theory, and transcendence. Proc. ICM 1, 476–483 (1986)
Zucker, S.: The Hodge conjecture for cubic fourfolds. Compositio Math. 34(2), 199–209 (1977)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2830-9_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2829-3
Online ISBN: 978-1-4939-2830-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)