×

zbMATH — the first resource for mathematics

Jumps in the Archimedean height. (English) Zbl 07108020
Summary: We introduce a pairing on local intersection cohomology groups of variations of pure Hodge structure, which we call the asymptotic height pairing. Our original application of this pairing was to answer a question on the Ceresa cycle posed by R. Hain and D. Reed. (This question has since been answered independently by Hain.) Here we show that a certain analytic line bundle, called the biextension line bundle, and defined in terms of normal functions, always extends to any smooth partial compactification of the base. We then show that the asymptotic height pairing on intersection cohomology governs the extension of the natural metric on this line bundle studied by Hain and Reed (as well as, more recently, by several other authors). We also prove a positivity property of the asymptotic height pairing, which generalizes the results of a recent preprint of J. Burgos Gil, D. Holmes and R. de Jong, along with a continuity property of the pairing in the normal function case. Moreover, we show that the asymptotic height pairing arises in a natural way from certain Mumford-Grothendieck biextensions associated to normal functions.
MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] O. Amini, S. J. Bloch, J. I. Burgos Gil, and J. Fresán, Feynman amplitudes and limits of heights, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 5, 5-40.
[2] D. Bertrand, Extensions panachées autoduales, J. K-Theory 11 (2013), no. 2, 393-411. · Zbl 1272.18007
[3] P. Brosnan, H. Fang, Z. Nie, and G. Pearlstein, Singularities of admissible normal functions, with appendix by N. Fakhruddin, Invent. Math. 177 (2009), no. 3, 599-629. · Zbl 1174.14009
[4] P. Brosnan and G. Pearlstein, On the algebraicity of the zero locus of an admissible normal function, Compos. Math. 149 (2013), no. 11, 1913-1962. · Zbl 1293.32019
[5] P. Brosnan and G. Pearlstein, The zero locus of an admissible normal function, Ann. of Math. (2) 170 (2009), no. 2, 883-897. · Zbl 1184.32004
[6] J. I. Burgos Gil, D. Holmes, and R. de Jong, Singularities of the biextension metric for families of abelian varieties, Forum Math. Sigma 6 (2018), e12. · Zbl 1402.14032
[7] J. I. Burgos Gil, D. Holmes, and R. de Jong, Positivity of the height jump divisor, Int. Math. Res. Not. IMRN 2017, art. ID rnx169.
[8] E. Cattani and A. Kaplan, “Degenerating variations of Hodge structure” in Actes du Colloque de Théorie de Hodge (Luminy, 1987), Astérisque 179-180, Soc. Math. France, Paris, 1989, 67-96. · Zbl 0705.14006
[9] E. Cattani, A. Kaplan, and W. Schmid, Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), no. 3, 457-535. · Zbl 0617.14005
[10] E. Cattani, A. Kaplan, and W. Schmid, \(L^2\) and intersection cohomologies for a polarizable variation of Hodge structure, Invent. Math. 87 (1987), no. 2, 217-252. · Zbl 0611.14006
[11] D. A. Lear, Extensions of normal functions and asymptotics of the height pairing, Ph.D. dissertation, University of Washington, Seattle, Wash., 1990.
[12] M. A. A. de Cataldo and L. Migliorini, On singularities of primitive cohomology classes, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3593-3600. · Zbl 1174.14011
[13] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, Berlin, 1970. · Zbl 0244.14004
[14] J. Giraud, Cohomologie non abélienne, Grundlehren Math. Wiss. 179, Springer, Berlin, 1971.
[15] H. Grauert and R. Remmert, Plurisubharmonische functionen in komplexen räumen, Math. Z. 65 (1957), 175-194. · Zbl 0070.30403
[16] M. Green and P. Griffiths, “Algebraic cycles and singularities of normal functions” in Algebraic Cycles and Motives, Vol. 1, London Math. Soc. Lecture Note Ser. 343, Cambridge Univ. Press, Cambridge, 2007, 206-263. · Zbl 1130.14010
[17] R. Hain, Biextensions and heights associated to curves of odd genus, Duke Math. J. 61 (1990), no. 3, 859-898. · Zbl 0737.14005
[18] R. Hain, “Normal functions and the geometry of moduli spaces of curves” in Handbook of Moduli, Vol. I, Adv. Lect. Math. (ALM) 24, Int. Press, Somerville, 2013, 527-578. · Zbl 1322.14049
[19] R. M. Hain, “Torelli groups and geometry of moduli spaces of curves” in Current Topics in Complex Algebraic Geometry (Berkeley, 1992/93), Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press, Cambridge, 1995, 97-143.
[20] R. M. Hain and D. Reed, On the Arakelov geometry of moduli spaces of curves, J. Differential Geom. 67 (2004), no. 2, 195-228. · Zbl 1118.14029
[21] C. Hardouin, Structure galoisienne des extensions itérées de modules différentiels, preprint, 2005, http://www.math.univ-toulouse.fr/ hardouin/thesclau2.pdf.
[22] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121-176. · Zbl 0431.14004
[23] T. Hayama and G. Pearlstein, Asymptotics of degenerations of mixed Hodge structures, Adv. Math. 273 (2015), 380-420. · Zbl 1345.58001
[24] D. Johnson, An abelian quotient of the mapping class group \(\mathcal{I}_g\), Math. Ann. 249 (1980), no. 3, 225-242. · Zbl 0409.57009
[25] D. L. Johnson. Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), no. 1, 119-125. · Zbl 0407.57003
[26] M. Kashiwara, A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci., 22 (1986), no. 5, 991-1024. · Zbl 0621.14007
[27] M. Kashiwara, \(D\)-modules and Microlocal Calculus, trans. by M. Saito, Transl. Math. Monogr. 217, Iwanami Series in Modern Mathematics, Amer. Math. Soc., Providence, 2003. · Zbl 1017.32012
[28] M. Kashiwara and T. Kawai. The Poincaré lemma for variations of polarized Hodge structure, Publ. Res. Inst. Math. Sci. 23 (1987), no. 2, 345-407. · Zbl 0629.14005
[29] K. Kato, C. Nakayama, and S. Usui, Classifying spaces of degenerating mixed hodge structures, IV: The fundamental diagram, Kyoto J. Math. 58 (2018), no. 2, 289-426. · Zbl 1423.14073
[30] K. Kato, C. Nakayama, and S. Usui, Classifying spaces of degenerating mixed Hodge structures, II: Spaces of \(\text{SL}(2)\)-orbits, Kyoto J. Math. 51 (2011), no. 1, 149-261. · Zbl 1233.14007
[31] M. Kerr and G. Pearlstein. Boundary components of Mumford-Tate domains, Duke Math. J., 165 (2016), no. 4, 661-721. · Zbl 1375.14045
[32] C. O. Kiselman, “Plurisubharmonic functions and potential theory in several complex variables” in Development of Mathematics 1950-2000, Birkhäuser, Basel, 2000, 655-714. · Zbl 0962.31001
[33] C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, with an appendix by S. I. Gelfand, corrected reprint, Mod. Birkhäuser Class., Birkhäuser, Basel, 2011. · Zbl 1237.14003
[34] G. J. Pearlstein, \(\text{SL}_2\)-orbits and degenerations of mixed Hodge structure, J. Differential Geom. 74 (2006), no. 1, 1-67.
[35] G. J. Pearlstein, Variations of mixed Hodge structure, Higgs fields, and quantum cohomology, Manuscripta Math. 102 (2000), no. 3, 269-310. · Zbl 0973.32008
[36] G. Pearlstein and C. Peters, Differential geometry of the mixed Hodge metric, to appear in Comm. Anal. Geom., preprint, arXiv:1407.4082 [math.AG]. · Zbl 1429.32045
[37] M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221-333. · Zbl 0727.14004
[38] M. Saito, Admissible normal functions, J. Algebraic Geom. 5 (1996), no. 2, 235-276. · Zbl 0918.14018
[39] W. Schmid. Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211-319. · Zbl 0278.14003
[40] J.-P. Serre. Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955-1956), 1-42. · Zbl 0075.30401
[41] J.-P. Serre. Prolongement de faisceaux analytiques cohérents, Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 1, 363-374. · Zbl 0144.08003
[42] A. Grothendieck, P. Deligne, and N. Katz, Groupes de monodromie en géométrie algébrique, I, Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I), Lecture Notes in Math. 288, Springer, Berlin, 1972.
[43] The Stacks Project, http://stacks.math.columbia.edu, 2016.
[44] J. H. M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math. 79 (1985), no. 3, 557-565. · Zbl 0568.14021
[45] J. Steenbrink and S. Zucker. Variation of mixed Hodge structure, I. Invent. Math. 80 (1985), no. 3, 489-542. · Zbl 0626.14007
[46] D. Treumann. Stacks similar to the stack of perverse sheaves, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5395-5409. · Zbl 1206.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.