Variation of Hodge structures and hypergeometric functions

在本研究阶段的开始，我们计划研究一组环面超曲面的临界值与拓扑环之间的关系。这种关系是完全清楚的。更具体地说，我们通过构造泛族环面超曲面在凸体上的紧化来研究它们。给出了次多面体模紧化上的一个显式紧族。利用这种构造，我们发现了与临界点相关的稳定循环与超几何函数的形式展开之间的对应关系。我们研究的另一个主题是关于复指数参数的Selberg积分的展开。我们在这个题目上得到了一个完整的结果。我们证明了展开式的系数表示为多个零值的线性组合。我们要考虑的微分形式是组合确定的，它恰好等于Falk-Terao后的β-nbc碱。我们期待这方面的进一步发展。

In the beginning of this research period, we are planning to study the relation between critical values and topological cycles for a family of toric hypersurfaces. This relateion is completely made clear. More concretely, we study them by constructing a compactification of the universal family of toric hypersurfaces in terms of convex bodies. We give an explicit comact family over the compactification of moduli associated to the secondary polytope. Using this construction, we found a correspondece between the stable cycles assciated to critical points and a formal expansion of hypergeometric functions.Another theme of our research is on the expansion of Selberg integral with resepect to the complex exponent parameter. We obtain a complete result on this subject. We proved that the coefficient of the expansion is expressed as a linear combination of multiple zera values. The differential forms that we should consider is determined combinatorially and it happens to be equal to β-nbc base after Falk-Terao. We expect further developement in this directions.