Restriction theorems: from orbits and Chevalley to periods and Galois
Bong Lian, Kamryn Spinelli
TL;DR
This paper develops a Galois-theoretic framework for restriction phenomena in invariant theory, generalizing Chevalley’s restriction theorem to arbitrary subvarieties via two restriction properties and reinterpreting Weyl groups as Galois groups of invariant-field extensions. It constructs an extension-parameter scheme $ ext{M}= ext{Spec}(oldmath C[V] ens_{oldmath C[V]^G}K)$ whose closed points parametrize extension subvarieties $Y_p$, linking their coordinate rings and function fields to a common ambient field $K$. By introducing the positive closure and establishing equivalences between algebraic and geometric formulations, it derives conditions under which Chevalley restriction holds and examines deformations of extension subvarieties through complex-parameter families. The latter machinery is then used to obtain explicit period formulas for Calabi–Yau families by lifting subfamily hypergeometric data to the full moduli space, with concrete results for CY double covers of projective spaces and elliptic curves in $oldsymbol P^2$, expressed via invariant functions on parameter spaces. Overall, the work bridges invariant theory, Galois theory, and period computations, offering a unified approach to mirror-symmetry period calculations and suggesting further avenues for lifting known subfamily formulas to global moduli via restriction properties.
Abstract
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.