论文标题
通过热带几何形状的超曲面的周期积分
Period integrals of hypersurfaces via tropical geometry
论文作者
论文摘要
令$ \ left \ {z_t \ right \} _ t $是一个复杂的尺寸的复杂超曲面的单参数，$ d \ geq 1 $在感谢您的多种多样。我们通过应用使用热带几何形状的abouzaid-andatra-iritani-sheridan来计算$ \ left \ {z_t \ right \} _ t $的周期积分的渐近学。作为集成媒体，我们考虑了Meromormorphic $（D+1）$的Poincaré残留物，形式在环境复合体上，它们沿Hypersurface $ Z_T $具有杆子。我们整合它们的周期是球形和托里，与热带$（0，d）$ - 循环和$（d，0）$ - 循环在$ \ weft \ left \ {z_t \ right \ right \} _ t $的热带化上。在$ d = 1 $的情况下，我们明确地写下Kato-usui的两极分化对数霍奇结构，以推论为限制。在本文中，我们施加了这样的假设，即热带化对牛顿多层人士的单型三角剖分是双重的。
Let $\left\{ Z_t \right\}_t$ be a one-parameter family of complex hypersurfaces of dimension $d \geq 1$ in a toric variety. We compute asymptotics of period integrals for $\left\{ Z_t \right\}_t$ by applying the method of Abouzaid--Ganatra--Iritani--Sheridan, which uses tropical geometry. As integrands, we consider Poincaré residues of meromorphic $(d+1)$-forms on the ambient toric variety, which have poles along the hypersurface $Z_t$. The cycles over which we integrate them are spheres and tori which correspond to tropical $(0, d)$-cycles and $(d, 0)$-cycles on the tropicalization of $\left\{ Z_t \right\}_t$ respectively. In the case of $d=1$, we explicitly write down the polarized logarithmic Hodge structure of Kato--Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.
