## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Analyticity of the closures of some Hodge theoretic subspaces

#### Abstract

In this paper, we prove a general theorem concerning the analyticity of the closure of a subspace defined by a family of variations of mixed Hodge structures, which includes the analyticity of the zero loci of degenerating normal functions. For the proof, we use a moduli of the valuative version of log mixed Hodge structures.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 9 (2011), 167-172.

Dates
First available in Project Euclid: 4 November 2011

https://projecteuclid.org/euclid.pja/1320417395

Digital Object Identifier
doi:10.3792/pjaa.87.167

Mathematical Reviews number (MathSciNet)
MR2863360

Zentralblatt MATH identifier
1252.14010

#### Citation

Kato, Kazuya; Nakayama, Chikara; Usui, Sampei. Analyticity of the closures of some Hodge theoretic subspaces. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 9, 167--172. doi:10.3792/pjaa.87.167. https://projecteuclid.org/euclid.pja/1320417395

#### References

• P. Brosnan and G. Pearlstein, The zero locus of an admissible normal function, Ann. of Math. (2) 170 (2009), no. 2, 883–897.
• P. Brosnan and G. Pearlstein, Zero loci of admissible normal functions with torsion singularities, Duke Math. J. 150 (2009), no. 1, 77–100.
• P. Brosnan and G. Pearlstein, On the algebraicity of the zero locus of an admissible normal function, arXiv:0910.0628.
• P. Brosnan, G. Pearlstein and C. Schnell, The locus of Hodge classes in an admissible variation of mixed Hodge structure, C. R. Math. Acad. Sci. Paris 348 (2010), no. 11–12, 657–660.
• F. Charles, On the zero locus of normal functions and the étale Abel-Jacobi map, Int. Math. Res. Not. IMRN 2010, no. 12, 2283–2304.
• P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer, Berlin, 1970.
• P. A. Griffiths, Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties, Amer. J. Math. 90 (1968), 568–626.
• M. Kashiwara, A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci. 22 (1986), no. 5, 991–1024.
• K. Kato, C. Nakayama and S. Usui, Moduli of log mixed Hodge structures, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 7, 107–112.
• K. Kato, C. Nakayama and S. Usui, Néron models in log mixed Hodge theory by weak fans, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 8, 143–148.
• K. Kato, C. Nakayama and S. Usui, Classifying spaces of degenerating mixed Hodge structures, II: Spaces of $\mathrm{SL}(2)$-orbits, Kyoto J. Math. 51 (2011), no. 1, Nagata Memorial Issue, 149–261.
• K. Kato, C. Nakayama and S. Usui, Classifying spaces of degenerating mixed Hodge structures, III: Spaces of nilpotent orbits.
• K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures, Annals of Mathematics Studies, 169, Princeton Univ. Press, Princeton, NJ, 2009.
• J. D. Lewis, A filtration on the Chow groups of a complex projective variety, Compositio Math. 128 (2001), no. 3, 299–322.
• M. Saito, Admissible normal functions, J. Algebraic Geom. 5 (1996), no. 2, 235–276.
• M. Saito, Hausdorff property of the Néron models of Green, Griffiths and Kerr, arXiv:0803.2771.
• W. Schmid, Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211–319.
• C. Schnell, Complex analytic Néron models for arbitrary families of intermediate Jacobians, arXiv:0910.0662, to appear in Invent. Math.
• S. Usui, Variation of mixed Hodge structure arising from family of logarithmic deformations II: Classifying space, Duke Math. J. 51 (1984), no. 4, 851–875.