The paper under review is the first part of the elaboration of the authors’ innovating program to study the mirror symmetry phenomenon from the viewpoint of Fontaine-Illusie-Kato log-structures on degenerations of Calabi-Yau manifolds. This interesting new approach to mirror symmetry was announced in their report [Turk. J. Math. 27, No. 1, 33–60 (2003;Zbl 1063.14048)], back then as a survey on work in progress. In the meantime, the authors have made significant progress in carrying out this announced program, and the present first comprehensive paper is to lay the foundations of it in as detailed as possible a manner.
The authors’ new approach to mirror symmetry is based on their crucial observation that certain kinds of degenerations of Calabi-Yau manifolds allow to construct a natural log-structure on their degenerate fibre, and that this log Calabi-Yau space essentially encodes all the information concerning the mirror properties of the given degeneration itself. This is to say that virtually all mirror statements for the so-called “large complex structure limit” associated to that degeneration can already be derived from the degenerate fibre viewed as a log Calabi-Yau space. The fundamental ingredient of this approach is given by particular “residual data” associated to certain maximally unipotent degenerations \(f: X\to S\) of Calabi-Yau varieties, which basically consist of the central fibre \(X_0\) of the degeneration, equipped with the log-structure induced by the embedding \(X_0\subset X\), a polarization, and a certain element of an adjoint “log Kähler moduli space”. In this context, two degenerating families of Calabi-Yau varieties are expected to be mirror-dual if and only if their associated residual degeneration data are dual in a precisely defined sense.
With regard to this philosophy in the study of mirror symmetry, the most important result of the present paper is the construction of a fundamental duality between logarithmic complex moduli and Kähler moduli of central fibres of degenerations as log-spaces, while further and deeper consequences will be presented in the forthcoming second part of the paper under review.
The foundational toolkit for all these novel constructions is comprehensively developed in the five sections of the present paper, and that in an utmost detailed, systematic, rigorous, yes even monograph-like style of presentation.
Section 1 introduces affine manifolds and those structures on them which will play a significant role in the authors’ work as a whole. Affine manifolds with singularities, their combinatorial polyhedral decompositions, discrete Legendre transforms, positivity and simplicity properties, and further analogues to torus fibrations are the objects of study that lead to an algebro-geometric version of the Strominger-Yau-Zaslow approach to mirror symmetry via “\(T\)-duality”.
Section 2 begins the process of constructing toric log Calabi-Yau spaces (as algebraic spaces) from integral affine manifolds with singularities and fixed polyhedral decomposition, whereas Section 3 is devoted to log-structures, their sheaf theory, and their appearance in the just created framework of toric log Calabi-Yau spaces associated to affine manifolds. Section 4 completes the definition of toric degenerations of Calabi-Yau varieties, which are among the very basic new concepts within the authors’ approach, and provides the essential link to the log Calabi-Yau spaces constructed before. In this context, a crucial role is played by the so-called “dual intersection complex”, a construction that basically reverses the constructions carried out in the foregoing two sections.
Section 5 finally completes the picture by combining the previous constructions and results. The authors give a precise description of the moduli of log Calabi-Yau spaces with fixed dual intersection complex in a special (simple) case, showing that this moduli space coincides with a certain cohomology group of a sheaf determined canonically by the underlying affine manifold structure, and that this group is precisely the group expected from the previous analysis of the Strominger-Yau-Zaslow conjecture [cf.A. Strominger,S.-T. Yau andE.Zaslow, Nucl. Phys., B 479, No. 1–2, 243–259 (1996;Zbl 0896.14024)]. Finally, one of the main concepts in the sequel, that is in the forthcoming second part of the authors’ work, namely the log Kähler moduli space of a toric log Calabi-Yau space \(X\), is defined, and the preliminary main theorem on the duality between logarithmic complex moduli spaces and log Kähler moduli spaces for polarized log Calabi Yau spaces is concluded.
The authors end this paper with a further outlook to their current program, including open questions, goals, and their treatment in the sequel. Also, there is an appendix collecting some facts on complexes arising from polyhedral decompositions of affine manifolds, as they were used in Section 1. As the paper is partly rather technical, forced by the delicate matter, the authors have also added an index of notations for the reader’s convenience.
Undoubtedly, this is a highly original and pioneering approach to mirror symmetry, with the highlights being to be expected, in the sequel to the paper under review.

Reviewer: Werner Kleinert (Berlin)