The Hodge realization of the polylogarithm on the product of multiplicative groups
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Mathematische Zeitschrift
The Hodge realization of the polylogarithm on the product of multiplicative groups Kenichi Bannai1,2 · Kei Hagihara2 · Kazuki Yamada1 · Shuji Yamamoto1,2 Received: 27 July 2018 / Accepted: 28 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this article is to describe explicitly the polylogarithm class in absolute Hodge cohomology of a product of multiplicative groups, in terms of the Bloch–Wigner– Ramakrishnan polylogarithm functions. We will use the logarithmic Dolbeault complex defined by Burgos to calculate the corresponding absolute Hodge cohomology groups. Mathematics Subject Classification 14C30 · 11G55
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mixed Hodge structure on cohomology . . . . . . . . . . . . 2.1 Construction of mixed Hodge structures on cohomology . 2.2 Admissible unipotent variation of mixed Hodge structures 3 Geometric polylogarithm . . . . . . . . . . . . . . . . . . . 3.1 The logarithm sheaf . . . . . . . . . . . . . . . . . . . . 3.2 Cohomology of the logarithm sheaf . . . . . . . . . . . . 3.3 The geometric polylogarithm class . . . . . . . . . . . . 3.4 Explicit construction of the geometric polylogarithm . . 4 Logarithmic Dolbeault complex . . . . . . . . . . . . . . . . 4.1 Review of logarithmic Dolbeault complex . . . . . . . . 4.2 Cohomology of the logarithm sheaf . . . . . . . . . . . . 4.3 Mixed Hodge structure of the logarithm sheaf . . . . . . 5 Absolute polylogarithm . . . . . . . . . . . . . . . . . . . . 5.1 The absolute polylogarithm class . . . . . . . . . . . . .
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This research was conducted as part of the KiPAS program FY2014–2018 of the Faculty of Science and Technology at Keio University. This research was supported in part by KAKENHI 26247004, 16J01911, 16K13742, 18H05233 as well as the JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.
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Kenichi Bannai [email protected]
1
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan
2
Mathematical Science Team, RIKEN Center for Advanced Intelligence Project )AIP), 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
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K. Bannai et al. 5.2 Polylogarithm function and the Ccase g =
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