Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T14:16:26.046Z Has data issue: false hasContentIssue false
  • English
  • Français

On the algebraicity of the zero locus of an admissible normal function

Part of: Deformations of analytic structures Families, fibrations

Published online by Cambridge University Press:  28 August 2013

Patrick Brosnan  and
Gregory Pearlstein
Patrick Brosnan
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA email pbrosnan@umd.edu
Gregory Pearlstein
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA email gpearl@math.tamu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure.

Keywords

normal functionsHodge theory

MSC classification

Secondary: 32G20: Period matrices, variation of Hodge structure; degenerations 14D07: Variation of Hodge structures 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 11 , November 2013 , pp. 1913 - 1962
Copyright
© The Author(s) 2013 

References

Brosnan, P. and Pearlstein, G. J., The zero locus of an admissible normal function, Ann. of Math. (2) 170 (2009), 883–897.Google Scholar
Brosnan, P., Pearlstein, G. and Schnell, C., The locus of Hodge classes in an admissible variation of mixed Hodge structure, C. R. Math. Acad. Sci. Paris 348 (2010), 657–660.Google Scholar
Carlson, J. A., The geometry of the extension class of a mixed Hodge structure, in Algebraic geometry, Bowdoin 1985, Brunswick, Maine, 1985, Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 199–222.Google Scholar
Cattani, E. and Kaplan, A., Degenerating variations of Hodge structure, Astérisque 9 (1989), 67–96; Actes du Colloque de Théorie de Hodge (Luminy, 1987).Google Scholar
Cattani, E., Kaplan, A. and Schmid, W., Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), 457–535.CrossRefGoogle Scholar
Deligne, P., Théorie de Hodge. II, Publ. Math. Inst. Hautes Études Sci. (1971), 5–57.CrossRefGoogle Scholar
Deligne, P., La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci. (1980), 137–252.Google Scholar
Deligne, P., Letter to E. Cattani and A. Kaplan, 1993.Google Scholar
Deligne, P., Structures de Hodge mixtes réelles, in Motives, Seattle, WA, 1991, Proceedings of Symposia in Pure Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1994), 509–514.Google Scholar
Deligne, P., Milne, J. S., Ogus, A. and Shih, K.-y., Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, Berlin, 1982).Google Scholar
Hironaka, H., Bimeromorphic smoothing of a complex-analytic space, Acta Math. Vietnam. 2 (1977), 103–168.Google Scholar
Jacobson, N., Lie algebras (Dover Publications Inc., New York, 1979), (republication of the 1962 original).Google Scholar
Kaplan, A. and Pearlstein, G., Singularities of variations of mixed Hodge structure, Asian J. Math. 7 (2003), 307–336.Google Scholar
Kashiwara, M., A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci. 22 (1986), 991–1024.CrossRefGoogle Scholar
Kato, K., Nakayama, C. and Usui, S., $\mathrm{SL} (2)$-orbit theorem for degeneration of mixed Hodge structure, J. Algebraic Geom. 17 (2008), 401–479.CrossRefGoogle Scholar
Kato, K., Nakayama, C. and Usui, S., Moduli of log mixed Hodge structures, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 107–112.Google Scholar
Milne, J. S., Quotients of Tannakian categories, Theory Appl. Categ. 18 (2007), 654–664.Google Scholar
Pearlstein, G. J., Variations of mixed Hodge structure, Higgs fields, and quantum cohomology, Manuscripta Math. 102 (2000), 269–310.Google Scholar
Pearlstein, G., ${\mathrm{SL} }_{2} $-orbits and degenerations of mixed Hodge structure, J. Differential Geom. 74 (2006), 1–67.Google Scholar
Saito, M., Admissible normal functions, J. Algebraic Geom. 5 (1996), 235–276.Google Scholar
Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.Google Scholar
Schwarz, C., Relative monodromy weight filtrations, Math. Z. 236 (2001), 11–21.Google Scholar
Schnell, C., Complex analytic Néron models for arbitrary families of intermediate Jacobians, Invent. Math. 188 (2012), 1–81.CrossRefGoogle Scholar
Serre, J.-P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955–1956), 1–42.Google Scholar
Demazure, M. and Grothendieck, A.(eds), Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Mathematics, vol. 152 (Springer, Berlin, 1970).Google Scholar
Saavedra Rivano, N., Catégories Tannakiennes, Lecture Notes in Mathematics, vol. 265 (Springer, Berlin, 1972).CrossRefGoogle Scholar
Tits, J., Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220.Google Scholar
Ziegler, P., Graded and filtered fiber functors on Tannakian categories, Preprint (2011), arXiv:1111.1981.Google Scholar