Canonical Coordinates in Toric Degenerations

Abstract

We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods directly carry enumerative information with no further gauging necessary as opposed to the classical case. A side result is that the canonical formal families lift to analytic families. We compute the relevant period integrals explicitly. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the degenerate Calabi-Yau.

39 pages, 4 figures. This paper has been completely rewritten and with strengthened results is now contained in arXiv:1907.03794 [math.AG]

References in corpus (1)

• Local mirror symmetry in the tropics

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