## Kyoto Journal of Mathematics

### Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram

#### Abstract

We complete the construction of the fundamental diagram of various partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. The diagram includes the space of nilpotent orbits, the space of $\mathrm{SL}(2)$-orbits, and the space of Borel–Serre orbits. We give amplifications of this fundamental diagram and amplify the relations of these spaces. We describe how this work is useful in understanding asymptotic behaviors of Beilinson regulators and of local height pairings in degeneration. We discuss mild degenerations in which regulators converge.

#### Article information

Source
Kyoto J. Math., Volume 58, Number 2 (2018), 289-426.

Dates
Accepted: 23 September 2016
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.kjm/1513674221

Digital Object Identifier
doi:10.1215/21562261-2017-0024

Mathematical Reviews number (MathSciNet)
MR3799704

Zentralblatt MATH identifier
06896956

#### Citation

Kato, Kazuya; Nakayama, Chikara; Usui, Sampei. Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram. Kyoto J. Math. 58 (2018), no. 2, 289--426. doi:10.1215/21562261-2017-0024. https://projecteuclid.org/euclid.kjm/1513674221

#### References

• [1] A. A. Beilinson, “Higher regulators and values of L-functions” (in Russian) in Current Problems in Mathematics, Vol. 24, Itogi Nauki i Tekhniki. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, 181–238; English translation in J. Soviet Math. 30 (1985), 2036–2070.
• [2] A. A. Beilinson, “Height pairing between algebraic cycles” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, 1987, 1–24.
• [3] S. J. Bloch, Height pairings for algebraic cycles, J. Pure Appl. Algebra 34 (1984), 119–145.
• [4] S. J. Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, CRM Monogr. Ser. 11, Amer. Math. Sci., Providence, 2000.
• [5] S. Bloch and K. Kato, Asymptotic behaviors of heights and regulators in degeneration, in preparation.
• [6] A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491.
• [7] P. Brosnan and G. Pearlstein, On the algebraicity of the zero locus of an admissible normal functions, Compos. Math. 149 (2013), 1913–1962.
• [8] P. Brosnan and G. Pearlstein, Jumps in the Archimedean height, preprint, arXiv:1701.05527 [math.AG].
• [9] E. Cattani, A. Kaplan, and W. Schmid, Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), 457–535.
• [10] A. B. Goncharov and A. M. Levin, Zagier’s conjecture on $L(E,2)$, Invent. Math. 132 (1998), 393–432.
• [11] P. A. Griffiths, Periods of integrals on algebraic manifolds, I: Construction and properties of modular varieties, Amer. J. Math. 90 (1968), 568–626.
• [12] T. Hayama and G. Pearlstein, Asymptotics of degenerations of mixed Hodge structures, Adv. Math. 273 (2015), 380–420.
• [13] K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\mathbf{C}$, Kodai Math. J. 22 (1999), 161–186.
• [14] K. Kato, C. Nakayama, and S. Usui, $\mathrm{SL}(2)$-orbit theorem for degeneration of mixed Hodge structure, J. Algebraic Geom. 17 (2008), 401–479.
• [15] K. Kato, C. Nakayama, and S. Usui, Classifying spaces of degenerating mixed Hodge structures, I, Adv. Stud. Pure Math. 54 (2009), 187–222; II, Kyoto J. Math. 51 (2011), 149–261; III, J. Algebraic Geom. 22 (2013), 671–772.
• [16] K. Kato and S. Usui, “Borel–Serre spaces and spaces of SL(2)-orbits” in Algebraic Geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002, 321–382.
• [17] K. Kato and S. Usui, Classifying Spaces of Degenerating Polarized Hodge Structures, Ann. of Math. Stud. 169, Princeton Univ. Press, Princeton, 2009.
• [18] T. Oda, Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. (3) 15, Springer, Berlin, 1988.
• [19] G. Pearlstein, $\mathrm{SL}_{2}$-orbits and degenerations of mixed Hodge structure, J. Differential Geom. 74 (2006), 1–67.
• [20] N. Schappacher and A. J. Scholl, The boundary of Eisenstein symbol, Math. Ann. 290 (1991), 303–321.
• [21] W. Schmid, Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211–319.
• [22] S. Usui, Variation of mixed Hodge structure arising from family of logarithmic deformations, II: Classifying space, Duke Math. J. 51 (1984), 851–875.