Journal of the Ramanujan Mathematical Society

Volume 28A, Issue SPL, July – Special Issue 2013 pp. 353–408.

Algebro-geometric stability and crystals

Authors: Tadao Oda
Author institution: Professor Emeritus, Tohoku University

Summary: In this survey, the author reports the recent applications in discrete geometric analysis of his joint work with Seshadri in the 1970's on the compactifications of the generalized Jacobian variety of a nodal curve by means of Geometric Invariant Theory. The algebro-geometric stability was described in terms of the dual graph of the nodal curve. The description led to convex polyhedral facet-to-facet and space-filling tilings, called Namikawa tilings, of Euclidean spaces with respect to lattices and to the combinatorial description of the compactifications. The convex polyhedral facet-to-facet and space-filling Namikawa tilings generalizing those above and appearing in the more general orthonormal setting, not necessarily coming from graphs, turn out to be related to the “cut and project” method for crystals and quasicrystals. The crystals constructed as the standard realization of finite graphs by Kotani and Sunada turn out to have hidden in it a Voronoi tiling and a Namikawa tiling at least in the case of maximal abelian coverings of the graph. Guided by examples, the author poses a question and formulates a conjecture in the case of non-maximal free abelian coverings.

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