Variation of Hodge structures and hypergeometric functions

在本研究的开始阶段，我们计划研究一族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，我们通过构造关于凸体的环面超曲面族的紧化来研究它们。我们给出了与二次多面体相关的模紧化的显式紧化族。利用这种构造，我们找到了与临界点相关的稳定圈与超几何函数的形式展开式之间的对应关系。我们研究的另一个主题是Selberg积分对复指数参数的展开式。我们在这个问题上得到了一个完整的结果。我们证明了展开系数表示为多个Zera值的线性组合。我们应该考虑的微分形式是组合确定的，它恰好等于福尔克-特劳之后的β-Nbc基。我们期待着这一方向的进一步发展。

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.