Top

Published in:

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

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Introduction to Nonabelian Hodge Theory

next chapter Mirror Symmetry in Physics: The Basics

That is, its finite-dimensional representations are reducible.

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.

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.

We thank C. Robles for providing this example.

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.

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.)

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”.

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

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

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.

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

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

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).

Abdulali, S.: Hodge structures of CM type. J. Ramanujan Math. Soc. 20(2), 155–162 (2005)MathSciNetMATH

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)MathSciNetCrossRefMATH

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)MathSciNetMATH

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)MathSciNetCrossRefMATH

Brosnan, P., Pearlstein, G.: On the algebraicity of the zero locus of an admissible normal function. Compositio Math. 149, 1913–1962 (2013)MathSciNetCrossRefMATH

Brosnan, P., Fang, H., Nie, Z., Pearlstein, G.: Singularities of normal functions. Invent. Math. 177, 599–629 (2009)MathSciNetCrossRefMATH

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)CrossRef

10.

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)

11.

Cattani, E.: Mixed Hodge Structures, Compactifications and Monodromy Weight Filtration. Annals of Math Study, vol. 106, pp. 75–100. Princeton University Press, Princeton (1984)

12.

Cattani, E., Deligne, P., Kaplan, A.: On the locus of Hodge classes. JAMS 8, 483–506 (1995)MathSciNetMATH

13.

Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. 123, 457–535 (1986)MathSciNetCrossRefMATH

14.

Charles, F.: On the zero locus of normal functions and the étale Abel-Jacobi map. IMRN 2010(12), 2283–2304 (2010)MathSciNetMATH

15.

Clingher, A., Doran, C.: Modular invariants for lattice polarized K3 surfaces. Mich. Math. J. 55(2), 355–393 (2007)MathSciNetCrossRefMATH

16.

Cohen, P.: Humbert surfaces and transcendence properties of automorphic functions. Rocky Mt. J. Math. 26(3), 987–1001 (1996)CrossRefMATH

17.

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)

18.

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)CrossRef

19.

Dettweiler, M., Reiter, S.: Rigid local systems and motives of type G ₂. Compositio Math. 146, 929–963 (2010)MathSciNetCrossRefMATH

20.

Doran, C., Kerr, M.: Algebraic cycles and local quantum cohomology Commun. Number Theory Phys. 8, 703–727 (2014)MathSciNetCrossRefMATH

21.

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)

22.

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)

23.

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

24.

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)

25.

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)CrossRef

26.

Green, M., Griffiths, P., Kerr, M.: Néron models and limits of Abel-Jacobi mappings. Compositio Math. 146, 288–366 (2010)MathSciNetCrossRefMATH

27.

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)

28.

Griffiths, P., Robles, C., Toledo, D.: Quotients of non-classical flag domains are not algebraic (2013, preprint). Available at arXiv:1303.0252

29.

Hazama, F.: Hodge cycles on certain abelian varieties and powers of special surfaces. J. Fac. Sci. Univ. Tokyo Sect. 1a 31, 487–520 (1984)

30.

Holzapfel, R.-P.: The Ball and Some Hilbert Problems. Birkhäuser, Basel (1995)CrossRefMATH

31.

Iritani, H.: An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Adv. Math. 222(3), 1016–1079 (2009)MathSciNetCrossRefMATH

32.

Jefferson, R., Walcher, J.: Monodromy of inhomogeneous Picard-Fuchs equations (2013, preprint). arXiv:1309.0490

33.

Kato, K., Usui, S.: Classifying Spaces of Degenerating Polarized Hodge Structure. Annals of Mathematics Studies, vol. 169, Princeton University Press, Princeton (2009)

34.

Katz, N.: Rigid Local Systems. Princeton University Press, Princeton (1995)

35.

Kerr, M., Lewis, J.: The Abel-Jacobi map for higher Chow groups, II. Invent. Math. 170, 355–420 (2007)MathSciNetCrossRefMATH

36.

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)

37.

Kerr, M., Pearlstein, G.: Boundary Components of Mumford-Tate Domains. Duke Math. J. (to appear). arXiv:1210.5301

38.

Kerr, M., Pearlstein, G.: Naive boundary strata and nilpotent orbits. Ann. Inst. Fourier 64(6), 2659–2714 (2014)MathSciNetCrossRefMATH

39.

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

40.

Laporte, G., Walcher, J.: Monodromy of an inhomogeneous Picard-Fuchs equation. SIGMA 8, 056, 10 (2012)

41.

Lefschetz, S.: l’Analysis situs et la géometrié algébrique. Gauthier-Villars, Paris (1924)

42.

Movasati, H.: Modular-type functions attached to mirror quintic Calabi-Yau varieties (2012, preprint). arXiv:1111.0357

43.

Murty, V.K.: Exceptional Hodge classes on certain abelian varieties. Math. Ann. 268, 197–206 (1984)MathSciNetCrossRefMATH

44.

Patrikis, S.: Mumford-Tate groups of polarizable Hodge structures (2013, preprint). arXiv:1302.1803

45.

Peters, C., Steenbrink, J.: Mixed Hodge Structures. Ergebnisse der Mathematik Ser. 3, vol. 52. Springer, Berlin (2008)

46.

Robles, C.: Schubert varieties as variations of Hodge structure (2012, preprint). arXiv:1208.5453, to appear in Selecta Math

47.

Saito, M.: Hausdorff property of the Zucker extension at the monodromy invariant subspace (2008, preprint). arXiv:0803.2771

48.

Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)MathSciNetCrossRefMATH

49.

Schneider, T.: Einführung in die transzendenten Zahlen (German). Springer, Berlin (1957)CrossRef

50.

Schnell, C.: Two observations about normal functions. Clay Math. Proc. 9, 75–79 (2010)MathSciNet

51.

Schnell, C.: Complex-analytic Néron models for arbitrary families of intermediate Jacobians. Invent. Math. 188(1), 1–81 (2012)MathSciNetCrossRefMATH

52.

Shiga, H.: On the representation of the Picard modular function by θ constants I-II. Publ. RIMS Kyoto Univ. 24, 311–360 (1988)MathSciNetCrossRefMATH

53.

Shiga, H., Wolfart, J.: Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. 463, 1–25 (1995)MathSciNetMATH

54.

Sommese, A.: Criteria for quasi-projectivity. Math. Ann. 217, 247–256 (1975)MathSciNetCrossRefMATH

55.

Usui, S.: Generic Torelli theorem for quintic-mirror family. Proc. Jpn. Acad. Ser. A Math. Sci. 84(8), 143–146 (2008)MathSciNetCrossRefMATH

56.

Voisin, C.: Hodge loci and absolute Hodge classes. Compos. Math. 143(4), 945–958 (2007)MathSciNetMATH

57.

Wüstholz, G.: Algebraic groups, Hodge theory, and transcendence. Proc. ICM 1, 476–483 (1986)

58.

Zucker, S.: The Hodge conjecture for cubic fourfolds. Compositio Math. 34(2), 199–209 (1977)MathSciNetMATH

Title: Algebraic and Arithmetic Properties of Period Maps
Author: Matt Kerr
Publisher: Springer New York
Book: Calabi-Yau Varieties: Arithmetic, Geometry and Physics
Print ISBN: 978-1-4939-2829-3

Electronic ISBN: 978-1-4939-2830-9

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4939-2830-9_6

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner