We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber.

One application is a proof that the mirror map for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author is trivial. We also show that these families are the completion of an analytic family, without reparametrization, and that they are formally versal as deformations of logarithmic schemes. Other applications include canonical one-parameter type III degenerations of K3 surfaces with prescribed Picard groups.

As a technical result of independent interest we develop a theory of period integrals with logarithmic poles on finite order deformations of normal crossing analytic spaces.

我们给出了由壁结构和中心纤维与复曲面变体的结合构成的退化变体家族中规范全同体积形式的积分的简单表达。整合的循环由中心纤维交汇处的热带1循环构成。

一个应用程序证明了格罗斯（Gross）构造的卡拉比丘（Calabi-Yau）规范正规家庭的镜像图是微不足道的。我们还表明，这些族是分析族的完成，没有重新参数化，并且它们在形式上是对数形式的变形。其他应用包括具有规定的皮卡德基团的K3曲面的标准一参数III型退化。

作为具有独立利益的技术成果，我们开发了一种对数极点的周期积分理论，该周期积分是对法线交叉解析空间的有限阶变形。