限制定理：从轨道和Chevalley到周期和Galois
Restriction theorems: from orbits and Chevalley to periods and Galois
AI总结：
利用Galois理论的新方法，研究约化群复表示中满足不变环和函数域限制性质的子簇，并应用于Calabi-Yau族的周期积分显式公式。
摘要：
利用基于Galois理论的新方法，我们研究了约化群的复表示中满足其不变环和函数域限制性质的子簇，遵循Chevalley限制定理的思路。对于一类性质良好的表示，我们显式地参数化了这些限制性质的候选者，并解释了一种理解它们在复族中形变的技术。我们还给出了Chevalley限制性质的代数和几何刻画，阐明了这一视角如何与先前的轨道理论方法联系起来。最后，我们利用这些限制性质证明了某些Calabi-Yau族的周期积分的显式公式。关键洞察在于，函数域上的限制性质可用于在代数和解析设置之间进行局部插值。利用这一技术，我们从子族中提升超几何周期公式，得到了射影空间和$\mathbb{P}^2$中椭圆曲线的Calabi-Yau双覆盖的周期的新显式公式，这些公式用其参数空间上的不变函数表示。
英文摘要：
Using a new approach based on Galois theory, we study subvarieties of complex representations of reductive groups which satisfy restriction properties on their invariant rings and function fields, along the lines of the Chevalley restriction theorem. For a certain well-behaved class of representations, we explicitly parametrize candidates for these restriction properties and explain a technique to understand their deformations in complex families. We also give algebraic and geometric characterizations of the Chevalley restriction property which clarify how this perspective connects back to previous orbit-theoretic approaches. Finally, we utilize these restriction properties to prove explicit formulas for period integrals of some Calabi-Yau families. The key insight is that the restriction property on function fields can be leveraged to locally interpolate between the algebraic and analytic settings. Using this technique, we lift hypergeometric period formulas from subfamilies to obtain novel explicit formulas for periods of Calabi-Yau double covers of projective spaces and elliptic curves in $\mathbb{P}^2$, expressed in terms of invariant functions on their parameter spaces.
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