2019/07/08 by Helge Ruddat, Ruddat, Helge, Bernd Siebert +1 · 3 citations
Mathematics · Physics and Astronomy · #14H15 #14J32 #14J33 #14M25 #32G20 #32Q25 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1907.03794
openalex publication_date 2019/07/08 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28
We give a simple expression for the integral of the canonical holomorphic\nvolume form in degenerating families of varieties constructed from wall\nstructures and with central fiber a union of toric varieties. The cycles to\nintegrate over are constructed from tropical 1-cycles in the intersection\ncomplex of the central fiber. One application is a proof that the mirror map\nfor the canonical formal families of Calabi-Yau varieties constructed by Gross\nand the second author is trivial. We also show that these families are the\ncompletion of an analytic family, without reparametrization, and that they are\nformally versal as deformations of logarithmic schemes. Other applications\ninclude canonical one-parameter type III degenerations of K3 surfaces with\nprescribed Picard groups. As a technical result of independent interest we\ndevelop a theory of period integrals with logarithmic poles on finite order\ndeformations of normal crossing analytic spaces.\n