1 Introduction↩︎

Understanding the space of orbits of representation \(V\) of a reductive group \(G\) as an algebraic variety is a notoriously subtle task [1]. The natural idea of setting \(V \sslash G = \operatorname{mSpec}(\mathbb{C}[V]^G)\), or in other words defining the ring of functions on the quotient to agree with the ring of \(G\)-invariant functions on \(V\), leads only to a rough parametrization of the so-called semistable orbits whose closures do not contain the origin, and throws away lots of information about the unstable orbits. Furthermore, the fibers of the canonical morphism of algebraic varieties \(V \to V \sslash G\) are only truly orbits of the \(G\)-action above the stable locus of orbits which are closed and whose points have finite stabilizers. Algebraically complicating the situation is that understanding the invariant rings themselves is generally not straightforward, and highly involved computations may be necessary to determine the degrees of the generators of the invariant rings, let alone the explicit forms of these generators themselves. For example, the invariant ring of the ternary cubic is generated as a \(\mathbb{C}\)-algebra by the Aronhold invariants \(S\) and \(T\) whose explicit forms together occupy about a page and a half in [2], and the invariant ring of the ternary quartic is generated by thirteen elements, seven of which are algebraically independent, and which have degrees ranging from 3 to 27 [3], [4].

On the other hand, if one instead takes a birational perspective, there is always a geometric quotient. Namely, Rosenlicht showed that for any action of an algebraic group \(G\) on a variety \(X\), there is a variety \(Z\) whose function field is the field \(\mathbb{C}(X)^G\) of \(G\)-invariant rational functions, and such that the induced dominant rational map \(\tau : X \dashedrightarrow Z\) has \(G\)-orbits as its fibers [5]. Roughly, Rosenlicht’s theorem says that the invariant field is a finer algebraic object for parametrizing \(G\)-orbits, up to a Zariski-small subset of such orbits; the algebraic cost you pay is that the invariant field \(\mathbb{C}(X)^G\) contains the invariant ring \(\mathbb{C}[X]^G\) and perhaps other rational functions depending on the character theory of \(G\). Rosenlicht also proved that whenever \(G\) is a solvable algebraic group, there is a rational map \(\sigma : Z \dashedrightarrow X\) so that \(\tau \circ \sigma\) is the identity; in other words, the image of \(\sigma\) meets most orbits exactly once. Since most reductive groups occurring in nature are not solvable, this fact is not particularly suited for our case.

When \(G\) is reductive, however, Luna gave a similar result with the tradeoff that the stabilizers of points enter the picture. This is known as Luna’s slice theorem. Given a point \(x\) of an affine \(G\)-variety \(X\) with closed stabilizer \(\operatorname{stab}(x)\), he showed that there is an affine subvariety \(S\) containing \(x\) which is stable under \(\operatorname{stab}(x)\), a Zariski open set \(U \subseteq V\), and a natural étale map \(S \times_{\operatorname{stab}(x)} G \to U\). This result has the same spirit as Palais’s slice theorem in the smooth category, which says that for any point \(x\) in a smooth \(G\)-space \(M\), there is a submanifold \(S\) containing \(x\) for which \(M\) is locally diffeomorphic to \(S \times (G / \operatorname{stab}(x))\) near \(x\) [6].

Many results in invariant theory have been motivated essentially by the idea of finding a global slice for the \(G\)-action, or something sufficiently close to it. (In fact, Luna’s slice theorem plays a crucial role in the proofs of some of the results described below.) In practice, the idea has historically been to find a vector subspace \(Y \subset V\) and a finite group \(W\) acting linearly on \(Y\) such that the restriction of polynomials \(\operatorname{res}_{V \to Y}\) sends \(\mathbb{C}[V]^G\) isomorphically to \(\mathbb{C}[Y]^W\); then this reduces the question of determining the invariant ring under the reductive group action to the simpler question of determining the invariant ring under a finite group action. In this article, we will call a statement which guarantees the existence of such a \(Y\) and \(W\) a restriction theorem.

The first, and probably best-known, example of such a theorem is Chevalley’s restriction theorem [7]. It deals with the specific case of the adjoint representation on a semisimple Lie algebra. Namely, let \(G\) be a connected semisimple Lie group, \(\mathfrak{g}\) its Lie algebra, \(\mathfrak{h}\) a Cartan subalgebra, and \(W\) its Weyl group, and let \(G\) act on \(\mathfrak{g}\) by the adjoint action and \(W\) on \(\mathfrak{h}\) in the usual way. Then the Chevalley restriction theorem states that \(\operatorname{res}_{\mathfrak{g}\to \mathfrak{h}} : \mathbb{C}[\mathfrak{g}]^G \to \mathbb{C}[\mathfrak{h}]^W\) is an isomorphism of invariant rings.

For instance, take \(G = \operatorname{SL}_2(\mathbb{C})\). In this case \(\mathfrak{h}\) is one-dimensional and \(W = \mathbb{Z}/ 2\mathbb{Z}\) acts by reflection on \(\mathfrak{h}\). Denoting by \(z_1, z_2, z_3\) the coordinate functions on \(\mathfrak{g}\) with respect to the usual basis \(H, X, Y\) for \(\mathfrak{sl}_2(\mathbb{C})\) and taking \(\mathfrak{h}= \operatorname{span}_\mathbb{C}(H)\), we have \(\mathbb{C}[\mathfrak{h}]^W = \mathbb{C}[z_1^2]\) and therefore \(\mathbb{C}[\mathfrak{g}]^G\) is generated by a single polynomial which restricts to \(z_1^2\) on \(\mathfrak{h}\). By manual computations involving the \(G\)-action on \(\mathfrak{g}\), one can find that \(\mathbb{C}[\mathfrak{g}]^G\) is generated by \(z_1^2 + z_2 z_3\), which is Killing dual to the Casimir element of \(\mathfrak{sl}_2(\mathbb{C})\) and is readily verified to be \(G\)-invariant.

Chevalley’s theorem caught the eye of many invariant theorists, who subsequently worked to generalize it to the context of reductive group actions. One such program was the study of polar representations by Dadok and Kac [8], [9]. In their orbit-theoretic approach, they studied vector subspaces of a reductive group representation \(V\) of the form \[\mathfrak{c}_v = \{x \in V \ \big|\ \mathfrak{g}\cdot x \subseteq \mathfrak{g}\cdot v\}\] whose \(G\)-orbit through each point has a tangent space parallel to the \(G\)-orbit through a given \(v \in V\). Dadok and Kac proved that when \(V\) is polar, meaning that some such subspace has the same dimension as \(V \sslash G\), it satisfies a restriction theorem. In this case \(\mathfrak{c}_v\) is called a Cartan subspace of \(V\) and there is a finite group \(W\), called the Weyl group, which is defined via a normalizer-mod-stabilizer construction inside \(G\) and acts on \(\mathfrak{c}_v\), for which \(\operatorname{res}_{V \to \mathfrak{c}_v} : \mathbb{C}[V]^G \to \mathbb{C}[\mathfrak{c}_v]^W\) is an isomorphism.

All of the adjoint representations addressed by the Chevalley restriction theorem are polar, and Dadok and Kac further enumerated all polar representations of simple reductive groups. A simple example would be the action of \(G = \operatorname{SL}_2(\mathbb{C})\) on \(V = M_{2 \times 2}(\mathbb{C})\), the space of \(2 \times 2\) complex matrices. The subspace \[\mathfrak{c}_{\operatorname{Id}} = \left\{ \begin{bmatrix} x & 0 \\ 0 & x \end{bmatrix} {\;\Bigg |\;} x \in \mathbb{C}\right\}\] is thus a Cartan subspace, and the Weyl group \(W = \mathbb{Z}/ 2 \mathbb{Z}\) acts on \(\mathfrak{c}_{\operatorname{Id}}\) by inversion. It is well-known that in this case \(\mathbb{C}[V]^G = \mathbb{C}[\Delta]\), where \(\Delta\) denotes the determinant, and one can easily check that its restriction to \(\mathfrak{c}_{\operatorname{Id}}\), \(x^2\), generates \(\mathbb{C}[\mathfrak{c}_{\operatorname{Id}}]^W\). We will see that the theory of the present article gives a new interpretation of Dadok and Kac’s Weyl group as a Galois group of a suitable field extension.

Another restriction theorem generalizing the work of Chevalley is due to Luna and Richardson [10]. They showed that in fact any representation \(V\) of a reductive group \(G\) satisfies a weak kind of restriction theorem. Namely, for generic \(x \in V\), there is a group \(W\) defined by a normalizer-mod-stabilizer construction in \(G\) which acts on the fixed point set \(F = V^{\operatorname{stab}x}\) under the action of the stabilizer of \(x\). In this case, \(\operatorname{res}_{V \to F} : \mathbb{C}[V]^G \to \mathbb{C}[F]^W\) is an isomorphism.

The keen reader will notice that it’s not guaranteed that \(W\) is finite; even in simple examples, it is not uncommon for \(W\) to be infinite if stabilizers of points in \(V\) are generically small. The case of \(\operatorname{SL}_2(\mathbb{C})\) acting on \(M_{2 \times 2}(\mathbb{C})\) is an extreme example: any invertible matrix has trivial stabilizer, and so the output of the Luna-Richardson theorem is \(F = M_{2 \times 2}(\mathbb{C})\) and \(W = \operatorname{SL}_2(\mathbb{C})\). However, when the group \(W\) is finite, the theorem can be used to deduce information on the invariant ring \(\mathbb{C}[V]^G\), and Luna and Richardson used this technique to compute the degrees of the invariant polynomials of some high-dimensional representations of \(\operatorname{SL}_7(\mathbb{C})\), \(\operatorname{SL}_8(\mathbb{C})\), and \(\operatorname{Spin}_{13}(\mathbb{C})\).

Despite the obvious power of these restriction theorems, some important questions still remain unanswered. For one, taking a geometric rather than representation-theoretic point of view, there is no obvious reason to require the target subvariety of a restriction theorem to be a vector subspace (or even that the ambient \(V\) must be a vector space). Furthermore, in each of the three restriction theorems above, the construction of the Weyl group is somewhat ad-hoc, and sorely lacks a satisfying uniform description. Our goals in this article are to clarify these points, and in particular we will explain that (1) these restriction theorems have a natural extension to arbitrary subvarieties of \(V\), not just subspaces; (2) the notion of restriction on function fields is a more natural condition to study from an algebraic perspective, especially through the lens of Galois theory, which plays a crucial role in restriction phenomena and can be used to universally define the Weyl group; and (3) that these insights are of profound and previously unnoticed utility in mirror symmetry and the study of period integrals of Calabi-Yau families. The purpose of this article is to explain a new approach to understanding restriction theorems which is rooted in Galois theory and to showcase novel insights to period integrals which follow from it.

Periods of Calabi-Yau (CY) families are a central object of study in mirror symmetry, as the local Torelli theorem [11], [12] essentially implies that they dictate the “B-model" information in the mirror correspondence. As a result, periods are known to encode enumerative [13], [14] and arithmetic [15]–[17] information about CY families. While periods of many CY families are known to be governed by GKZ generalized hypergeometric systems [18]–[21], the best-understood examples occur in non-canonically chosen gauge-fixed subfamilies of larger universal families. An illustrative example is to contrast the Weierstrass family of elliptic curves \(y^2 = x^3 + ax^2 + bx + c\), for which the period \(\int_\gamma \frac{dx}{y}\) is classically and explicitly known to be closely related to the Gauss hypergeometric function \(\leftindex_2{F}_1\), with the cubic family of elliptic curves given as hypersurfaces in \(\mathbb{P}^2\), for which there is also an explicit formula thanks to the theory of GKZ hypergeometric equations [22] and their general solutions [19].

In order to explicitly understand the periods of larger, universal families of CY varieties such as the full cubic family in \(\mathbb{P}^2\), one needs a procedure to lift the periods from a low-dimensional subvariety to the entire moduli space. A first step in this direction was taken by Lian and Yau [23], who derived a Poincaré residue method to canonically normalize the volume forms on Calabi-Yau complete intersections in a fixed compact manifold \(X\). From this canonical normalization, they constructed certain \(\mathcal{D}\)-modules, called tautological systems, specifically tailored to govern the periods. Whether or not the period sheaf exactly coincides with the solution sheaf of the tautological system depends on the cohomology of \(X\) and a choice of extra equivariance data baked into the tautological system, but is known to be the case in all examples we will consider in this article [24]–[26]. Our key insight on periods in this article is that this extra equivariance data can be leveraged using invariant theory to lift hypergeometric period formulas from low-dimensional subfamilies to the entire moduli space.

1.1 Structure of the article↩︎

The structure of this article is as follows. In Section 2, we will define the Chevalley restriction property, which captures the essence of previous restriction theorems, and the Galois restriction property, which generalizes the former to the level of function fields. These properties will be the main focus of this article.

In Section 3, we focus on the case of representations whose \(G\)-invariant rational functions are fractions of \(G\)-invariant polynomials. Such representations are sometimes called visible (see [27]), though this adjective also has other common meanings. In this setting, we construct a scheme \(\mathcal{M}\) over a large algebraically closed field \(K\) whose closed points parametrize so-called extension subvarieties, which are candidates for the Galois and Chevalley restriction properties. A key point is that the construction canonically embeds the coordinate rings and function fields of all extension subvarieties into \(K\), which allows us to phrase the relationship between the Galois and Chevalley restriction properties in terms of an intersection condition on subrings of \(K\).

Then, in Section 4, we continue under the assumptions of the previous section and push the question of the relationship between the Chevalley and Galois restriction properties. Motivated by the intersection condition in the previous section, we define the positive closure of an integral domain \(R\), a subset of \(\overline{\operatorname{Frac}R}\) which is closely intertwined with the Chevalley restriction property. Under suitable conditions on \(R\), we interpret the positive closure geometrically as a weak kind of surjectivity condition, which leads to the following equivalent characterizations of the Chevalley restriction property.

As an easy corollary of the Theorem, we recover a statement similar to a useful proposition of Gatti and Viniberghi (aliases of Kac and Vinberg), which highlights from an orbit-theoretic point of view why the Chevalley restriction property is a true generalization of previous restriction theorems.

Section 5 deals with the question of how to understand deformations of extension subvarieties. Using a technique reminiscent of Weil restriction, we construct spaces (more precisely, constructible sets) whose points are in bijection with points of \(\mathcal{M}\) with coordinates belonging to a given \(\mathbb{C}\)-sub-vector space of \(K\), and organize the corresponding extension subvarieties as a family over this base. We primarily focus on examples and open questions; this appears to be a ripe setting to investigate the properties of “generic" extension subvarieties.

Finally, in Section 6, we show the relevance of restriction properties to the study of period integrals and B-model moduli for CY families. We begin by proving two short lemmas which interpolate between the algebraic and analytic settings when given a subvariety with the Galois restriction property. In some sense, these lemmas have the flavor of a local slice theorem in the analytic category. We then prove striking explicit formulae for periods of CY double covers of \(\mathbb{P}^n\) and elliptic curves in \(\mathbb{P}^2\) in terms of invariant functions on their respective parameter spaces.

The history and the present article can be summarized in the following flowchart. Here, an arrow indicates inspiration or generalization. A solid box or arrow indicates a previous result and a dashed box or arrow indicates an insight introduced in this article.

Although we hope that the reader will be convinced that invariant theory and period integrals have much to offer each other, we see this article as just the beginning of the story. Throughout the text, we provide numerous examples and point out questions for future study. We hope that this will alert others to the many interesting questions waiting to be answered in this new area.

1.2 Notations and conventions.↩︎

In this article, all reductive groups are algebraic groups over \(\mathbb{C}\) and all their representations are \(\mathbb{C}\)-vector spaces. If \(X\) is a variety over a field \(k\), we use \(k[X]\) to denote the coordinate ring of \(X\) and \(\widehat{X} = \operatorname{Spec}k[X]\) to mean the scheme-theoretic analogue of \(X\). We use \(\mathbb{C}[V]^G\) to denote the subring of \(\mathbb{C}[V]\) comprising \(G\)-invariant functions, and \(\mathbb{C}(V)\) and \(\mathbb{C}(V)^G\) to mean the function field of \(V\) and its subfield of \(G\)-invariant functions. For \(f \in k[X]\), we use \(\mathcal{V}(f)\) to mean the vanishing locus of \(f\) in the variety \(X\). In Sections 3 and 4 only, we take \(V\) to be a representation satisfying \(\mathbb{C}(V)^G = \operatorname{Frac}(\mathbb{C}[V]^G)\). \(\operatorname{Frac}R\) denotes the field of fractions of an integral domain \(R\). We use the adjective “factorial" to mean that \(R\) is a UFD. Finally, if \(R\) is a reduced finitely-generated \(k\)-algebra, we use \(\operatorname{mSpec}R\) to mean the variety of closed points of \(\operatorname{Spec}R\).

2 Two restriction properties↩︎

In this section, we will push the results of Chevalley, Dadok-Kac, and Luna-Richardson further by defining two restriction properties that generalize the historical results outlined in Section 1. Given a representation \(V\) of a reductive group \(G\), we are interested in restriction properties where \(G\)-invariants on \(V\) restrict to invariant functions on a subvariety \(Y\) under the action of a finite group \(W\). Our first definition, on the level of invariant rings, aligns closely with the previous results in Section 1.

In this language, the Chevalley restriction theorem says that any Cartan subalgebra of \(\mathfrak{g}\) has the Chevalley restriction property, and the Dadok-Kac theorem says that any Cartan subspace of a polar representation has the Chevalley restriction property. However, there are plenty of cases not covered by these two theorems. In the example of \(\operatorname{SL}_2(\mathbb{C})\) acting on \(M_{2 \times 2}(\mathbb{C})\), the subvariety \(Y_1 = \{z_{12} = z_{21} = 0, z_{22} = 1\}\) is not a subspace, but has the Chevalley restriction property with \(W_1\) the trivial group. The subvariety \(Y_2 = \{z_{12} = z_{21} = 0, z_{11} = z_{22}^2\}\) is not even a degree-one subvariety, but has the Chevalley restriction property under the action of \(W_2 = \mathbb{Z}/ 3\mathbb{Z}\) by \((z_{11}, z_{22}) \mapsto (\zeta^2 z_{11}, \zeta z_{22})\), where \(\zeta\) is a third root of unity.

As we will see later, restriction theorems on function fields play an important role in the study of period integrals, motivating our second definition.

Note that in the setting of this definition, the extension \(\mathbb{C}(Y) / \mathbb{C}(Y)^W\) is automatically Galois, which justifies the terminology. A question which will occupy much of our attention is the precise relationship between these two restriction properties; the following is a first result in this direction.

Again in the example of \(\operatorname{SL}_2(\mathbb{C})\) acting on \(M_{2 \times 2}(\mathbb{C})\), now consider the subvariety \(Y_3 = \{z_{12} = z_{21} = 0, z_{11} z_{22}^2 = 1\}\). Here, \(\mathbb{C}[Y_3] \cong \frac{\mathbb{C}[z_{11}, z_{22}]}{(z_{11} z_{22}^2 - 1)} \cong \mathbb{C}[z_{22}, z_{22}^{-1}]\) and \(\mathbb{C}(Y_3) = \operatorname{Frac}\mathbb{C}[Y] \cong \mathbb{C}(z_{22})\). Taking \(W_3\) to be the trivial group, the restriction of the determinant \(\Delta\) which generates \(\mathbb{C}(V)^G\) to \(\mathbb{C}(Y_3)^{W_3} = \mathbb{C}(Y_3)\) is the generator \(z_{22}^{-1}\). However, on the level of invariant rings, \(\operatorname{res}_{V \to Y} : \mathbb{C}[V]^G \to \mathbb{C}[Y]^{W_3}\) is not an isomorphism: it is not surjective because its image does not contain any positive powers of \(z_{22}\). Therefore this is a simple example of a subvariety which has the Galois restriction property, but not the Chevalley restriction property.

3 Parametrizing extension subvarieties↩︎

In this section, we will take \(V\) to be a representation of a reductive group \(G\) with the property that \(\mathbb{C}(V)^G = \operatorname{Frac}(\mathbb{C}[V]^G)\). This condition is automatic in the case where \(G\) is connected and semisimple. We set \(\widehat{V} = \operatorname{Spec}\mathbb{C}[V]\) and \(K = \overline{\mathbb{C}(V)^G}\). We denote \(\widehat{V}_K = \widehat{V} \times_{\operatorname{Spec}\mathbb{C}} \operatorname{Spec}K\) the base change of \(\widehat{V}\) to \(K\), and \(\pi : \widehat{V}_K \to \widehat{V}\) the induced map.

Based on Proposition 1, it seems plausible that it should be possible to upgrade the Galois restriction property to the Chevalley restriction property under certain conditions. We will see that in order to understand the interplay between these two restriction properties, we need to canonically embed \(\mathbb{C}[V]^G\) and \(\mathbb{C}[Y]\) in a common ambient ring. This motivates the following definition.

\(\mathcal{M}\) is nonempty because \(\mathbb{C}[V]\) and \(K\) are nontrivial torsion-free \(\mathbb{C}[V]^G\)-modules. In coordinates, given a generating set \(\{P_1, \dots, P_m\}\) of \(\mathbb{C}[V]^G\), \(\mathcal{M}\) has the explicit description \[\mathcal{M}= \operatorname{Spec}\left( \frac{\mathbb{C}[V] \otimes_\mathbb{C}K}{(P_1 \otimes 1 - 1 \otimes P_1, \dots, P_m \otimes 1 - 1 \otimes P_m)} \right).\]

For the rest of this section, let \(p\) be a closed point of \(\mathcal{M}\).

The extension parameter scheme has two main uses: to parametrize extension subvarieties of \(V\) in geometric terms, and to canonically embed their coordinate rings into \(K\). In Section 5, the canonical embedding of coordinate rings into \(K\) will be used again to understand deformations of extension subvarieties.

The proof of Proposition 2 shows that the restriction maps \(\operatorname{res}_{V \to Y_p}\) sending \(\mathbb{C}[V]^G \to \mathbb{C}[Y_p]\) and \(\mathbb{C}(V)^G\) to \(\mathbb{C}(Y_p)\) are well-defined and injective: they are precisely the canonical inclusions of \(\mathbb{C}[V]^G\) into \(\mathbb{C}[Y_p]\) and \(\mathbb{C}(V)^G\) into \(\mathbb{C}(Y_p)\) described in the proposition. This allows us to show that the extension subvarieties are good candidates for restriction properties.

Because our construction canonically embeds the coordinate rings and function fields of extension subvarieties into \(K\), we can compare \(\mathbb{C}(V)^G\) and \(\mathbb{C}[Y]\) as subrings of \(K\) to relate the Chevalley and Galois restriction properties.

To illustrate this proposition, consider the final example from Section 2: in the representation \(M_{2 \times 2}(\mathbb{C})\) of \(\operatorname{SL}_2(\mathbb{C})\), consider the subvariety \(Y_3 = \{z_{12} = z_{21} = 0, z_{11} z_{22}^2 = 1\}\). Inside of \(K = \overline{\mathbb{C}(\Delta)}\), \(\mathbb{C}[Y_3] \cap \mathbb{C}(V)^G = \mathbb{C}[\Delta, \Delta^{-1}] \cap \mathbb{C}(\Delta)\). This intersection equals \(\mathbb{C}[\Delta, \Delta^{-1}]\), which is strictly larger than \(\mathbb{C}[V]^G\).

We conclude this section by enunciating the precise relationship between closed points in \(\mathcal{M}\) and extension subvarieties in \(V\).

4 The Chevalley restriction property in algebraic terms↩︎

Thanks to Proposition 4, the question of when a subvariety with the Galois extension property also has the Chevalley restriction property is reduced to a question of computing intersections of subrings of \(K\). Motivated by this perspective, we define the notion of the positive closure.

If \(R\) is a field, then \(\operatorname{PC}(R)\) is its algebraic closure, but explicitly computing \(\operatorname{PC}(R)\) is generally quite difficult. The relationship of this definition to the Chevalley restriction property is the following.

When \(R = \mathbb{C}[V]^G\) (again under the assumption that \(\mathbb{C}(V)^G = \operatorname{Frac}(\mathbb{C}[V]^G)\)) and \(S = \mathbb{C}[Y]\) where \(Y\) is a subvariety with the Galois restriction property, then Propositions 4 and 5 together imply that \(Y\) has the Chevalley restriction property iff \(\mathbb{C}[Y] \subseteq \operatorname{PC}(\mathbb{C}[V]^G)\).

The rest of this section will focus on understanding some properties and characterizations of the positive closure. Our first result is a criterion for testing whether a given element of \(\overline{\operatorname{Frac}R}\) belongs to \(\operatorname{PC}(R)\) and resembles the definition of an integral element of a ring extension.

Like the integral closure of \(R\) in \(\overline{\operatorname{Frac}R}\), the positive closure is transitive in the following sense.

However, in contrast to the integral closure, \(\operatorname{PC}(R)\) may not be a ring. We know of no criterion to tell when \(\operatorname{PC}(R)\) is a ring, except the simple observation above when \(R\) is a field, and any such condition applicable to the Chevalley restriction property would likely be exceedingly strict. For example, even the polynomial ring \(R = \mathbb{C}[x]\) fails to have a ring as its positive closure: take \(s_1, s_2\) to be the roots of \(f(y) = (x+1) y^2 - xy + 1\) in \(\overline{\mathbb{C}(x)}\); then \(s_1\) and \(s_2\) both belong to \(\operatorname{PC}(R)\), but their sum \(\frac{x}{x+1}\) does not. Even explicitly describing the positive closure of a simple example like \(R = \mathbb{C}[x]\) seems quite difficult.

Because \(\operatorname{PC}(R)\) is not a ring, in order to check whether \(Y\) has the Chevalley restriction property, it is not enough to just check that the generators of \(\mathbb{C}[V]\) map into \(\operatorname{PC}(\mathbb{C}[V]^G)\) under the map \(\mathbb{C}[V] \to \mathbb{C}[Y] \to K\). One main takeaway from this is that the Chevalley restriction property is rather poorly behaved from the algebraic viewpoint and consequently very difficult to detect in families of extension subvarieties. We will see evidence in Section 5 that the Galois restriction property is much more well-behaved in this regard.

In light of the above, we switch perspectives now to pursue a geometric formulation of the positive closure. It’s easy to see that the natural morphism from any extension subvariety \(Y\) to \(V \sslash G\) is dominant, but the Chevalley restriction property is in fact related to the following stronger notion.

Because of our intention of studying invariant rings, we will work in the case when \(R\) is a reduced factorial finitely-generated \(\mathbb{C}\)-algebra, for example a polynomial ring over \(\mathbb{C}\). (The factoriality assumption can probably be weakened.)

This geometric understanding of the positive closure now allows us to recharacterize the Chevalley restriction property.

Corollary 2 bears much resemblance to an important result of Gatti and Viniberghi; compare with Proposition 8 of [29]. Their work is a key ingredient in the work of Dadok and Kac on polar representations, who prove their restriction theorem by recognizing that every Cartan subspace intersects all closed \(G\)-orbits and only closed \(G\)-orbits. We have traded Gatti and Viniberghi’s hypothesis that \(Y\) only intersects closed \(G\)-orbits for the Galois restriction property, and our statement deals with an arbitrary finite group \(W\) arising as the Galois group of \(\mathbb{C}(Y) / \mathbb{C}(V)^G\) whereas their group \(W\) is required to be exhibited as a stabilizer of \(Y\). While Gatti and Viniberghi clearly recognized the importance of the invariant field in restriction theorems – see their Proposition 5 and the proof of their Proposition 8 – it appears that the role of Galois theory has gone unnoticed.

5 Complex-parameter families of extension subvarieties↩︎

Considering the extension subvarieties of \(V\) parametrized by the closed points of the \(K\)-scheme \(\mathcal{M}\), it seems natural to ask questions about the properties of a “generic" extension subvariety. However, \(\mathcal{M}\) appears too large to yield useful information on such genericity questions because any dense open in \(\mathcal{M}\) will parametrize extension subvarieties whose function fields cover \(K\). In this section, we will present examples of finite-dimensional complex families of extension subvarieties which illustrate the types of questions we are interested in.

Viewing the variety of closed points of \(\mathcal{M}\) as a closed subvariety of \(K^n\), our objective is cut \(\mathcal{M}\) down to a finite-dimensional complex space by considering its intersection with \(A^n\), where \(A\) is a finite-dimensional \(\mathbb{C}\)-sub-vector space of \(K\). The obstruction is that \(\mathcal{M}\cap A^n\) does not make sense as a scheme-theoretic object because \(\mathcal{M}\) is a \(K\)-scheme. To accomplish this goal, we need a strategy for passing between \(\mathbb{C}\)-schemes and \(K\)-schemes. The method we present below can be regarded as an analogue of Weil restriction (see [30], or [31] for a more classical description by Weil himself) and will be the subject of future work.

On the level of rings, the closed points we are interested in are precisely those for which the image of the \(\mathbb{C}\)-linear map \(V^* \to \mathbb{C}[V] \to \mathbb{C}[V] \otimes_{\mathbb{C}[V]^G} K \to K\) is contained in \(A\). Without loss of generality we may assume that the \(\mathbb{C}\)-subalgebra \(\langle A \rangle\) of \(K\) generated by the elements of \(A\) contains \(\mathbb{C}[V]^G\); if not, there are no such closed points. As such linear maps are elements of \(\operatorname{Hom}_\mathbb{C}(V^*, A) \cong V \otimes_\mathbb{C}A\), the space \(\mathcal{B}\) will be a subset of the scheme-theoretic analog \(\widehat{V \otimes_\mathbb{C}A}\) of this vector space. To view \(\widehat{V \otimes_\mathbb{C}A}\) as living inside of \(K^n\), we need to base change to \(K\), upon which we have a natural morphism of \(K\)-schemes \(\mu : (\widehat{V \otimes_\mathbb{C}A})_K \to \widehat{V}_K\) given on closed points by \(v \otimes a \mapsto a v\).

Thus, we may consider the inverse image \(\mu^{-1}(\mathcal{M}) = (\widehat{V \otimes_\mathbb{C}A})_K \times_{\widehat{V}_K} \mathcal{M}\). Suppose that \(\mu^{-1}(\mathcal{M})\) is cut out by an ideal \(I\) of \(\mathbb{C}[V \otimes_\mathbb{C}A] \otimes_\mathbb{C}K\) whose generators \(f_i\) are expressed in terms of elements \(a_i\) of \(\langle A \rangle\) and generators \(x_i\) of \(\mathbb{C}[V \otimes_\mathbb{C}A]\). From the geometric perspective, the \(a_i\) represent regular functions on \(S = \operatorname{Spec}\langle A \rangle\) and the \(f_i\) represent algebraic combinations of these algebraic functions over \(\mathbb{C}\) once complex values of the \(x_i\) have been chosen.

Thus, passing back from this \(K\)-scheme to a \(\mathbb{C}\)-scheme amounts to understanding the set of possible values of the \(x_i\) over \(\mathbb{C}\) that make the elements defining \(I\) identically zero. To do this, we take the scheme-theoretic image \(\mathcal{A}\) of the morphism \(\varphi : \mu^{-1}(\mathcal{M}) \to (\widehat{V \otimes_\mathbb{C}A}) \times_\mathbb{C}S\) given on the level of rings by \[\begin{align} \mathbb{C}[V \otimes_\mathbb{C}A] \otimes_\mathbb{C}\langle A \rangle &\to (\mathbb{C}[V \otimes_\mathbb{C}A] \otimes_\mathbb{C}K) / I \\ r \otimes s &\mapsto (r \otimes s) + I. \end{align}\] Geometrically, \(\mathcal{A}\) is the common zero locus of the \(f_i\) in \((\widehat{V \otimes_\mathbb{C}A}) \times_\mathbb{C}S\). It then follows that the set of \(x_i\) which make the \(f_i\) identically zero on \(S\) is the locus in \(\widehat{V \otimes_\mathbb{C}A}\) over which the fibers of \(\mathcal{A} \to \widehat{V \otimes_\mathbb{C}A}\) are the complete \(S\), i.e. for which the fibers are of full dimension \(\dim S\). By [32], this is a constructible set in \(\widehat{V \otimes_\mathbb{C}A}\) which we call \(\mathcal{B}\) (though in all examples we present \(\mathcal{B}\) will be an honest closed subvariety), and its closed points are in natural bijection with the closed points of \(\mathcal{M}\) lying in \(A^n\).

By a similar argument, the scheme-theoretic image \(\mathcal{X}\) of the morphism \(\psi : \mu^{-1}(\mathcal{M}) \to (\widehat{V \otimes_\mathbb{C}A}) \times_\mathbb{C}\widehat{V}\) comes with a natural map \(\rho : \mathcal{X} \to \widehat{V \otimes_\mathbb{C}A}\) with the property that \(\rho^{-1}(q)\) is precisely the scheme-theoretic image of \(\mu(q_K)\) under the fiber product map \(\pi : \widehat{V}_K \to \widehat{V}\). Therefore the fiber of \(\mathcal{X}\) over a point belonging to \(\mathcal{B} \subseteq \widehat{V \otimes_\mathbb{C}A}\) is the extension subvariety associated to the corresponding point of \(\mathcal{M}\). We can therefore view \(\mathcal{Y} = \rho^{-1}(\mathcal{B}) \to \mathcal{B}\) as the family of extension subvarieties indexed by closed points in \(\mathcal{M} \cap A^n\).

The rest of this section comprises examples of this technique used to construct families of extension subvarieties. Throughout, we use \(\operatorname{span}_\mathbb{C}(k_1, \dots, k_r)\) to mean the \(\mathbb{C}\)-sub-vector space of \(K\) spanned by \(k_1, \dots, k_r\).

These examples raise interesting questions about deformations of extension subvarieties, which will be the subject of a forthcoming article. Namely, in some fixed family \(\mathcal{Y} \to \mathcal{B}\) one can ask questions such as the following:

• What is the locus in \(\mathcal{B}\) above which the fibers all have the Galois restriction property? Under what conditions does the generic fiber have this property?

• Fix a Galois extension \(F\) of \(\mathbb{C}(V)^G\) inside of \(K\). Can we describe the locus in \(\mathcal{B}\) above which the canonical images inside \(K\) of the fibers’ function fields are equal to \(F\)? Is one option of \(F\) generic in the family?

• Perform this construction for each \(\mathbb{C}\)-sub-vector space \(A\) of \(K\) to get a family \(\mathcal{Y}_A \to \mathcal{B}_A\). Taking the direct limit of \(\mathcal{Y}_A \to \mathcal{B}_A\) over all such \(A\), we obtain a family of ind-constructible sets over \(\mathbb{C}\) which is some sense the “universal" family of extension subvarieties of \(V\). What properties does this universal family enjoy?

• The universal family in the previous bullet comes equipped with an action of the absolute Galois group \(\operatorname{Gal}(K / \mathbb{C}(V)^G)\). Can we say anything interesting about this action?

6 Period integrals from the Galois restriction property↩︎

In this section, we will demonstrate that restriction properties, especially the Galois restriction property on the level of function fields, are quite useful for understanding the period integrals of Calabi-Yau (CY) families. Specifically, we will leverage the Galois restriction property to write explicit formulas for the periods of CY families in terms of invariant functions.

In [23], Lian and Yau studied the period integrals of CY varieties \(Y\) given as complete intersections in an ambient manifold \(X\). Naively, these period integrals are defined by \[\Pi = \int_\gamma \omega_Y\] where \(\gamma \in H_{\dim Y}(Y, \mathbb{Z})\) and \(\omega_Y\) is a holomorphic volume form on \(Y\), which is unique up to scalar thanks to the fact that \(Y\) is CY.

The study of these period integrals is rather subtle: the parameter space of such complete intersections is a Zariski open set in a representation \(V\) of a reductive group \(G\) and is generally noncompact and noncontractible, so a canonical choice of volume form must be made on each complete intersection. Lian and Yau’s contribution was twofold: one, to provide a canonical normalization through a Poincaré residue technique on a special type of \(G\)-equivariant principal bundle over \(X\); and two, to describe a \(\mathcal{D}\)-module, called a tautological system, which governs the period integrals as functions on the moduli space. The construction can also be adapted to so-called “fractional complete intersections", which are CY branched covers of a fixed \(X\).

For simplicity, let \(X\) be a compact complex manifold whose anticanonical bundle \(-K_X\) is very ample. Consider the family of CY hypersurfaces given by the zero locus of sections of \(-K_X\), and let \(V = H^0(-K_X)^*\). We are interested in the locus of sections \(\sigma\) so that \(\{\sigma = 0\}\) is a smooth hypersurface; this forms a Zariski open set \(\mathcal{B}\) in \(V^*\) on which the tautological system is defined. There is an action of \(G = \operatorname{Aut}(X) \times \mathbb{C}^\times\) on \(\mathcal{B}\) induced by automorphisms of \(X\) and the \(\mathbb{C}^\times\)-action on scaling sections of \(-K_X\).

From this data, the tautological system is a \(\mathcal{D}\)-module comprising two types of operators: embedding polynomial operators encoding the geometry of the embedding of the ambient space \(X\) into \(\mathbb{P}V\), and symmetry operators encoding the transformation properties of the periods under the action of \(G\). In many cases, such as the case of hypersurfaces in \(\mathbb{P}^n\) or other homogeneous spaces, \(\operatorname{Aut}(X)\) is semisimple and the transformation properties encoded by the symmetry operators are quite close to \(G\)-invariance, typically taking the form of invariance under the action of \(\operatorname{Aut}(X)\) and homogeneity under the action of \(\mathbb{C}^\times\). Our objective is to combine this observation and the theory of extension subvarieties and their restriction properties, in order to lift solutions to the tautological system defined on subvarieties of \(\mathcal{B}\) to the entire moduli space. Although there are many descriptions of period integrals which can be viewed as functions on certain subvarieties of \(\mathcal{B}\), we are not aware of any procedure for finding explicit solutions to tautological systems in terms of \(G\)-invariants other than the method we are about to introduce. The technique relies on interplay between the algebraic and analytic categories, as explained in the two following lemmas.

Let \(V\) be any representation of a reductive group \(G\). (In particular, \(\mathbb{C}(V)^G\) may be strictly larger than \(\operatorname{Frac}(\mathbb{C}[V]^G)\).) By Rosenlicht’s theorem [5], there is a variety \(Z\) whose function field is \(\mathbb{C}(V)^G\) and a dominant rational map \(\pi : V \dashedrightarrow Z\) whose fibers are \(G\)-orbits in \(V\).

To deduce our invariant-theoretic period formulas, we will use Lemmas 4 and 5 to build a function on \(V\) with prescribed values on a subvariety and transformation properties under the \(\widehat{G}\)-action, then utilize the following lemma to argue that it must in fact be a period.

Throughout the discussion of the following period integral problems, we will make extensive use of work by Matsumoto, Sasaki, and Yoshida regarding reduction of hypergeometric systems to a particular closed subvariety. This result is really quite remarkable, because in general the question of restricting PDE systems or \(\mathcal{D}\)-modules to a subvariety is quite complex. We anticipate that the Galois restriction property plays a crucial role in this reduction process, and we plan to investigate it more deeply in the future.

6.1 Double covers of \(\mathbb{P}^{n-1}\)↩︎

We begin by considering the family of CY double covers of \(\mathbb{P}^{n-1}\) branched along \(2n\) hyperplanes in general position. Such a double cover is specified by a generic choice of \(2n\) sections of \(\mathcal{O}(1)\), or by a \(n \times 2n\) complex matrix \(Z = [z_{ij}]\) with no zero minors; call the space of such matrices by \(M_{n \times 2n}^o(\mathbb{C})\). This is a Zariski open subvariety of the representation \(M_{n \times 2n}(\mathbb{C})\) of \(G = \operatorname{SL}_n(\mathbb{C}) \times (\mathbb{C}^\times)^{2n}\), where \(\operatorname{SL}_n(\mathbb{C})\) acts by left multiplication on \(Z\) and the \(i\)-th copy of \(\mathbb{C}^\times\) scales the \(i\)-th column of \(Z\). Geometrically, the \(\operatorname{SL}_n(\mathbb{C})\) action is the action on \(\bigoplus_{i=1}^{2n} H^0(\mathcal{O}(1))\) induced by automorphisms of \(\mathbb{P}^{n-1}\), while the torus action simply rescales the sections defining the branch locus, whence it is easy to see that the group action identifies isomorphic double covers and preserves \(M_{n \times 2n}^o(\mathbb{C})\).

Adapting the Poincaré residue procedure defined by Lian and Yau for Calabi-Yau complete intersections [23] to the case of branched covers, one obtains the canonical volume form \[\omega(Z) = \frac{\sum_{i=1}^n (-1)^i w_i dw_1 \wedge \cdots \wedge \widehat{dw_i} \wedge \cdots \wedge dw_n}{\sqrt{\prod_{k=1}^{2n} z_{1,k} w_1 + \cdots + z_{n, k} w_n}}\] on each double cover, where \(w_1, \dots, w_n\) are the homogeneous coordinates on \(\mathbb{P}^{n-1}\) and the hat denotes omission. The resulting period \(\Pi(Z) = \int_\gamma \omega(Z)\) is a solution to the tautological system generated by the operators \[\begin{cases} \partial_{z_{ip}} \partial_{z_{jq}} - \partial_{z_{iq}} \partial_{z_{jp}} & 1 \leq i < j \leq n,\;1 \leq p < q \leq 2n \\ \sum_{j=1}^{2n} z_{ij} \partial_{z_{mj}} + \delta_{im} & 1 \leq i, m \leq n \\ \sum_{i=1}^n z_{ij} \partial_{z_{ij}} + \frac{1}{2} & 1 \leq j \leq 2n. \end{cases}\] The operators on the second line express the \(\operatorname{SL}_n(\mathbb{C})\)-invariance of the periods, while the operators on the third line express the fact that the periods are homogeneous of degree \(-\frac{1}{2}\) with regard to each \(\mathbb{C}^\times\) action, as can be seen from the expression for \(\omega(Z)\).

The tautological system for this family of double covers is a special case of the Aomoto-Gelfand hypergeometric system \(E(k, n)\) studied by Matsumoto, Sasaki, and Yoshida in [34] for the case \(k = 3\), \(n = 6\) and [35] for general \(k\) and \(n\). This is a PDE system on \(M_{k \times n}^o(\mathbb{C})\) with an action by \(G = \operatorname{SL}_k(\mathbb{C}) \times (\mathbb{C}^\times)^n\). In the latter article, they proved three key statements about this system and the geometry of the parameter space:

1. The subvariety \(Y \subset M_{k \times n}^o(\mathbb{C})\) consisting of matrices of the form \[\begin{bmatrix} 1 & & & & 1 & 1 & \ldots & 1 \\ & 1 & & & 1 & x_{2,k+2} & \ldots & x_{2,n} \\ & & \ddots & & \vdots & \vdots & \ddots & \vdots \\ & & & 1 & 1 & x_{k,k+2} & \ldots & x_{k,n} \end{bmatrix}\] intersects each \(G\)-orbit exactly once. In other words, the set \(G \cdot Y\) comprising \(G\)-translates of \(Y\) is all of \(M_{k \times n}^o(\mathbb{C})\).

2. The system \(E(k, n)\) has a reduction to a PDE system \(E'(k, n)\) on \(Y\) generated by second-order operators, in the sense that the restriction of a solution of \(E(k, n)\) to \(Y\) is a solution of \(E'(k, n)\).

3. There is an explicit power series solution of \(E'(k, n)\), unique up to scalar, which is holomorphic in a neighborhood of \[\begin{bmatrix} 1 & & & & 1 & 1 & \ldots & 1 \\ & 1 & & & 1 & 0 & \ldots & 0 \\ & & \ddots & & \vdots & \vdots & \ddots & \vdots \\ & & & 1 & 1 & 0 & \ldots & 0 \end{bmatrix}\] in \(Y\) and extends to a multivalued meromorphic function on \(Y\).

(Matsumoto, Sasaki, and Yoshida considered an action by the slightly larger group \(\operatorname{GL}_k(\mathbb{C})\;\times (\mathbb{C}^\times)^n\), but every \(M \in \operatorname{GL}_k(\mathbb{C})\) factors as the product of \((\det M)^{-1/k} M \in \operatorname{SL}_k(\mathbb{C})\) and \((\det M)^{1/k} I_k\), the latter of which acts on \(M_{k \times n}^o(\mathbb{C})\) identically to \(((\det M)^{1/k}, \dots, (\det M)^{1/k}) \in (\mathbb{C}^\times)^n\).) Our goal will be to lift the explicit solution of \(E'(n, 2n)\) from \(Y\) to a solution of \(E(n, 2n)\) on all of \(M_{n \times 2n}^o(\mathbb{C})\). Because we will work with rational functions, the restriction to a Zariski open makes no difference, and we can consider rational functions on all of \(M_{n \times 2n}(\mathbb{C})\). To do so, we must first understand the invariant theory of \(M_{n \times 2n}(\mathbb{C})\) as a representation of \(G\).

Note that in the case \(n = 2\), there is only one cross-ratio as defined in the Lemma, \(f_{24}\), which is exactly the familiar cross-ratio of four points on \(\mathbb{P}^1\). This cross-ratio is well-known as the only projective invariant of four points. In the language of invariant theory, Lemma 8 is saying that \(Y\) is a subvariety of \(M_{n \times 2n}(\mathbb{C})\) with the Galois restriction property, and \(\mathbb{C}(Y) / \mathbb{C}(M_{n \times 2n}(\mathbb{C}))^{G}\) is the trivial extension. For our purposes, this means that the map \(\rho\) defined in the analytic mapping lemma, Lemma 4, is single-valued and well-defined on the entire Zariski open \(M_{n \times 2n}^o\), sending \[Z \mapsto \begin{bmatrix} 1 & & & & 1 & 1 & \ldots & 1 \\ & 1 & & & 1 & f_{2,n+2}(Z) & \ldots & f_{2,2n}(Z) \\ & & \ddots & & \vdots & \vdots & \ddots & \vdots \\ & & & 1 & 1 & f_{n,n+2}(Z) & \ldots & f_{n,2n}(Z) \end{bmatrix}.\] Now we will leverage this map to apply the analytic lifting lemma, Lemma 5, in order to produce a period formula on all of \(M_{n \times 2n}^o\).

6.1.1 Example: elliptic curves as double covers of \(\mathbb{P}^1\)↩︎

In the case \(n = 2\), the family of elliptic curves given as CY double covers of \(\mathbb{P}^1\) is called the Legendre family. In fact, every elliptic curve arises in this way. Beginning with the action of \(G = \operatorname{SL}_2(\mathbb{C}) \times (\mathbb{C}^\times)^4\) on the parameter space \(M_{2 \times 4}^o(\mathbb{C})\), it is well-known that the classical cross-ratio of four points in \(\mathbb{P}^1\), given in terms of the matrix entries by \(\frac{\Delta_{14} \Delta_{23}}{\Delta_{13} \Delta_{24}}\), generates \(\mathbb{C}(M_{2 \times 4}(\mathbb{C}))^{G}\). From this, it is clear to see that the subvariety \(Y = \{\begin{bsmallmatrix} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & \lambda \end{bsmallmatrix}\}\) has the Galois restriction property with trivial Galois group, because the restriction of the cross-ratio to \(Y\) is exactly the coordinate function \(\lambda\). Geometrically, \(Y\) describes the family of elliptic curves cut out by the cubic equation \(y^2 = x (x-1) (x - \lambda)\). On \(Y\), the reduced Aomoto-Gelfand system \(E'(2, 4)\) is given by the single operator \[\theta^2 - \lambda \left( \theta + \frac{1}{2} \right)^2\] where \(\theta = \lambda \partial_\lambda\). This is the Gauss hypergeometric equation, and the Picard-Fuchs equation of this one-parameter subfamily described in [21]. The explicit solution to the reduced Aomoto-Gelfand system from [35] in this case is the Gauss hypergeometric function \[\varphi(\lambda) = \sum_{n=0}^\infty \frac{\Gamma(n + \frac{1}{2})^2}{\Gamma(\frac{1}{2})^2 \Gamma(n+1)^2} \lambda^n = \leftindex_2{F}_1 \left( \frac{1}{2}, \frac{1}{2}; 1; \lambda \right)\] which was also recovered in [21]. The \(\operatorname{SL}_2(\mathbb{C})\)-invariant and \((\mathbb{C}^\times)^4\)-homogeneous prefactor described above takes the form \[P(Z) = (\Delta_{13} \Delta_{24})^{-\frac{1}{2}}\] in this case. Putting the pieces together, we conclude that the holomorphic period of the Legendre family is the function on \(M_{2 \times 4}^o(\mathbb{C})\) defined by \[\Pi(Z) = (\Delta_{13} \Delta_{24})^{-1/2} \sum_{n=0}^\infty \frac{\Gamma\left( n + \frac{1}{2} \right)^2}{\Gamma\left( \frac{1}{2} \right)^2 \Gamma(n + 1)^2} \left( \frac{\Delta_{14} \Delta_{23}}{\Delta_{13} \Delta_{24}} \right)^n\] on a neighborhood where this power series converges, and extended to the rest of \(M_{2 \times 4}^o(\mathbb{C})\) by multivalued analytic continuation.

6.1.2 Example: K3 surfaces of Picard number 16 as double covers of \(\mathbb{P}^2\)↩︎

In the case \(n = 3\), the double covers of \(\mathbb{P}^2\) branched along six lines are K3 surfaces with \(A_1\) singularities above the intersections of pairs of lines. Blowing up the singularities yields a family of smooth K3 surfaces of Picard number 16. Using the action of \(\operatorname{SL}_3(\mathbb{C}) \times (\mathbb{C}^\times)^6\) on \(M_{3 \times 6}^o(\mathbb{C})\), every \(3 \times 6\) matrix with nonzero minors can uniquely be put into the form \[\begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & x_{25} & x_{26} \\ 0 & 0 & 1 & 1 & x_{35} & x_{36} \end{bmatrix},\] a process sometimes referred to as gauge fixing [20]. The subvariety of \(M_{3 \times 6}(\mathbb{C})\) comprising such matrices is the \(Y\) identified by Matsumoto, Sasaki, and Yoshida [35], and has the Galois restriction property with trivial Galois group. Namely, the coordinates \(x_{25}\), \(x_{35}\), \(x_{26}\), and \(x_{36}\) on \(Y\) are the restrictions of the \(G\)-invariant rational functions \[\frac{\Delta_{135} \Delta_{234}}{\Delta_{235} \Delta_{134}},\;\frac{\Delta_{125} \Delta_{234}}{\Delta_{235} \Delta_{124}},\;\frac{\Delta_{136} \Delta_{234}}{\Delta_{236} \Delta_{134}},\;\text{and}\;\frac{\Delta_{126} \Delta_{234}}{\Delta_{236} \Delta_{124}}\] on \(M_{3 \times 6}(\mathbb{C})\), respectively. Compactifications of \(Y\) were also studied in [21] in order to identify large complex structure limits of this family. The power series solution to the reduced Aomoto-Gelfand system \(E(3, 6)\) is \[\varphi(X) = \sum_N \frac{\Gamma(n_{25} + n_{35} + \frac{1}{2}) \Gamma(n_{26} + n_{36} + \frac{1}{2}) \Gamma(n_{25} + n_{26} + \frac{1}{2}) \Gamma(n_{35} + n_{36} + \frac{1}{2})}{2 \Gamma(\frac{1}{2})^3 \Gamma(n_{25} + n_{35} + n_{26} + n_{36} + \frac{3}{2}) \Gamma(n_{25} + 1) \Gamma(n_{35} + 1) \Gamma(n_{26} + 1) \Gamma(n_{36} + 1)} x_{25}^{n_{25}} x_{35}^{n_{35}} x_{26}^{n_{26}} x_{36}^{n_{36}}\] where the sum ranges over \(2 \times 2\) matrices \(N = \begin{bsmallmatrix} n_{25} & n_{26} \\ n_{35} & n_{36} \end{bsmallmatrix}\) with entries in \(\mathbb{Z}_{\geq 0}\), and the \(\operatorname{SL}_3(\mathbb{C})\)-invariant and \((\mathbb{C}^\times)^6\)-homogeneous prefactor for this family is \[P(Z) = \left( \frac{\Delta_{235} \Delta_{236} \Delta_{124} \Delta_{134}}{\Delta_{123} \Delta_{234}} \right)^{-\frac{1}{2}}.\] Hence, the holomorphic period of this family of K3 surfaces is given explicitly by \[\begin{align} \Pi(Z) &= \left( \frac{\Delta_{235} \Delta_{236} \Delta_{124} \Delta_{134}}{\Delta_{123} \Delta_{234}} \right)^{-\frac{1}{2}} \\ &\quad \times \sum_N {\Bigg [} \frac{\Gamma(n_{25} + n_{35} + \frac{1}{2}) \Gamma(n_{26} + n_{36} + \frac{1}{2}) \Gamma(n_{25} + n_{26} + \frac{1}{2}) \Gamma(n_{35} + n_{36} + \frac{1}{2})}{2 \Gamma(\frac{1}{2})^3 \Gamma(n_{25} + n_{35} + n_{26} + n_{36} + \frac{3}{2}) \Gamma(n_{25} + 1) \Gamma(n_{35} + 1) \Gamma(n_{26} + 1) \Gamma(n_{36} + 1)} \\ &\qquad \qquad \quad \times \left( \frac{\Delta_{135} \Delta_{234}}{\Delta_{235} \Delta_{134}} \right)^{n_{25}} \left( \frac{\Delta_{125} \Delta_{234}}{\Delta_{235} \Delta_{124}} \right)^{n_{35}} \left( \frac{\Delta_{136} \Delta_{234}}{\Delta_{236} \Delta_{134}} \right)^{n_{26}} \left( \frac{\Delta_{126} \Delta_{234}}{\Delta_{236} \Delta_{124}} \right)^{n_{36}} {\Bigg ]} \end{align}\] on the domain of convergence of this power series, and is defined on the rest of \(M_{3 \times 6}^o(\mathbb{C})\) by multivalued analytic continuation.

6.2 Elliptic curves in \(\mathbb{P}^2\)↩︎

Next, we consider the cubic family of elliptic curves, expressed as smooth hypersurfaces in \(\mathbb{P}^2\). These hypersurfaces are given by a nondegenerate section of \(\mathcal{O}(3)\), which can be written as a cubic in the homogeneous coordinates \(x_1\), \(x_2\), \(x_3\) of \(\mathbb{P}^2\) of the form \[\sigma = z_{111} x_1^3 + z_{112} x_1^2 x_2 + z_{113} x_1^2 x_3 + z_{122} x_1 x_2^2 + z_{123} x_1 x_2 x_3 + z_{133} x_1 x_3^2 + z_{222} x_2^3 + z_{223} x_2^2 x_3 + z_{233} x_2 x_3^2 + z_{333} x_3^3.\] The collection of such nondegenerate cubics is a Zariski open set \(V^o\) in the representation \(V = H^0(\mathcal{O}(3))\) of \(G = \operatorname{GL}_3(\mathbb{C})\), where the action of \(G\) is induced by its action on \(\mathbb{P}^2\).

In this scenario, the canonical volume form on the hypersurface \(\{\sigma = 0\}\) is given in terms of the coordinates \(z_{ijk}\) by \[\omega(Z) = \frac{x_1 dx_2 \wedge dx_3 - x_2 dx_1 \wedge dx_3 + x_3 dx_1 \wedge dx_2}{z_{111} x_1^3 + z_{112} x_1^2 x_2 + \cdots + z_{333} x_3^3}\] by the Poincaré residue method of Lian and Yau [23]. Thus the period integrals \[\Pi(Z) = \int_\gamma \omega(Z)\] are \(\operatorname{SL}_3(\mathbb{C})\)-invariant and homogeneous of degree \(-3\) with respect to the diagonal copy of \(\mathbb{C}^\times \subset \operatorname{GL}_3(\mathbb{C})\).

We will once again lift a formula for the period integral on a closed subvariety to a global period formula on the entire parameter space using the Galois restriction property. Remarkably, we will see that the period is a power series involving terms expressible in radicals. The end goal is to leverage what we already know about the Legendre family of elliptic curves from the previous section, but first we must touch upon the invariant theory of the \(G\)-action on \(V\). Since \(\operatorname{SL}_3(\mathbb{C})\) has no nontrivial characters, we have \(\mathbb{C}(V)^{\operatorname{SL}_3(\mathbb{C})} = \operatorname{Frac}(\mathbb{C}[V]^{\operatorname{SL}_3(\mathbb{C})})\), and it is a classical result that this invariant ring is a polynomial ring generated by the Aronhold invariants \(S\) and \(T\) of degrees \(4\) and \(6\) respectively (see Section 1). The \(j\)-invariant of the elliptic curve corresponding to a nondegenerate cubic in \(V\) is given in terms of \(S\) and \(T\) by the rational function \(J = \frac{S^3}{T^2 - 4 S^3}\), which generates \(\mathbb{C}(V)^G\) as an extension of \(\mathbb{C}\).

Now, it is a well-known fact that every elliptic curve over \(\mathbb{C}\) can be put into Weierstrass form \(a x_1^3 + b x_1^2 x_3 + c x_1 x_3^2 + d x_3^3 - x_2^2 x_3 = 0\) using the action of \(\operatorname{GL}_3(\mathbb{C})\) introduced above [36]. From this form, it is then simple to put the elliptic curve into Legendre form \(x_1 (x_1 - x_3) (x_1 - \lambda x_3) - x_2^2 x_3 = x_1^3 + (-\lambda - 1) x_1^2 z_3 + \lambda x_1 x_3^2 - x_2^2 x_3 = 0\) by another \(\operatorname{GL}_3(\mathbb{C})\)-transformation. We take \(Y\) to be the subvariety of \(V\) comprising those Legendre-form cubics; explicitly, \(Y\) is the one-dimensional subvariety \[\{z_{111} = 1,\;z_{113} = -\lambda - 1,\;z_{133} = \lambda,\;z_{223} = -1,\;z_{112} = z_{122} = z_{123} = z_{222} = z_{233} = z_{333} = 0\}\] with coordinate \(\lambda\) and the set \(G \cdot Y\) comprising its \(G\)-translates is all of \(V^o\).

However, a key difference between the subvariety \(Y\) for this family and the subvariety \(Y_L \subset M_{2 \times 4}(\mathbb{C})\) for the Legendre family identified in the previous section is that, while \(Y_L\) contained exactly one point in each \(G\)-orbit of \(M_{2 \times 4}^o(\mathbb{C})\), the intersection of \(Y\) with a \(G\)-orbit in \(V\) is not a single point but in fact (generically) six points, with an action of the symmetric group \(S_3\). More precisely, if \(y \in Y\) is a point with \(\lambda = \lambda_0\), then \[(G \cdot y) \cap Y = \left\{ \lambda = \lambda_0,\;1 - \lambda_0,\;\frac{1}{\lambda_0},\;\frac{\lambda_0}{\lambda_0 - 1},\;\frac{1}{1 - \lambda_0},\;\frac{\lambda_0 - 1}{\lambda_0} \right\}\] and the action of \(S_3\) on this set is simply the action on the cross-ratio of a quadruple of (distinct) points in \(\mathbb{P}^1\) familiar from enumerative geometry and the study of moduli problems (see, for example, [37]). It can also be observed in the family of double covers of \(\mathbb{P}^1\) as follows. The cross-ratio of a quadruple of points in \(\mathbb{P}^1\) represented by a matrix in \(M_{2 \times 4}^o\) is the bottom right entry of its orbit’s representative in \(Y_L\). Consider a point \(\begin{bsmallmatrix} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & \lambda_0 \end{bsmallmatrix} \in Y_L\) and permute the first three columns by an element of \(S_3\); then the representative in \(Y_L\) of the orbit of this column-permuted matrix is precisely the corresponding element in the above set.

Now we will show that \(Y\) has the Galois restriction property, that is, that \(\operatorname{res}_{V \to Y} : \mathbb{C}(V)^G \to \mathbb{C}(Y)\) is a Galois extension. Under our choice of normalization for the Aronhold invariants, the restrictions of \(S\) and \(T\) to \(Y\) are \[\lambda^2 - \lambda + 1 \quad \text{and} \quad 2 \lambda^3 - 3 \lambda^2 - 3 \lambda + 2\] respectively, thus \[\operatorname{res}_{V \to Y}(J) = \frac{(\lambda^2 - \lambda + 1)^3}{-27 \lambda^2 (\lambda-1)^2}.\] Clearing denominators and abusing notation by denoting \(\operatorname{res}_{V \to Y} J\) simply as \(J\), we have the equation \[\lambda^6 - 3 \lambda^5 + (27J + 6) \lambda^4 + (-54J - 7) \lambda^3 + (27J + 6) \lambda^2 - 3 \lambda + 1 = 0\] showing that \(\mathbb{C}(Y) / \mathbb{C}(V)^G\) is a degree-six extension. It is easy to check that the expression \(\frac{(\lambda^2 - \lambda + 1)^3}{-27 \lambda^2 (\lambda-1)^2}\) is invariant under the \(W = S_3\)-action on \(Y\), therefore \(\mathbb{C}(V)^G \subseteq \mathbb{C}(Y)^W\). Since \(\mathbb{C}(Y) / \mathbb{C}(Y)^W\) is automatically Galois, as the base is the fixed field of a group action, and because the degree of \(\mathbb{C}(Y) / \mathbb{C}(V)^G\) and the order of the group \(W\) are equal, we conclude that \(\mathbb{C}(V)^G = \mathbb{C}(Y)^W\). In other words, \(\mathbb{C}(Y) / \mathbb{C}(V)^G\) is Galois with Galois group \(W = S_3\), and so \((Y, W)\) has the Galois restriction property.

The preceding lemma exhibits \(\lambda\) as a multivalued function of \(J\) on \(Y\). From the point of view of the analytic mapping lemma, Lemma 4, a generic \(G\)-orbit in \(V^o\) intersects \(Y\) in six points, each corresponding to a choice of a branch of this multivalued function. Fixing a choice of a branch in a neighborhood of \(p \in V^o\) defines the map \(\rho\) of the analytic mapping lemma.

Under the canonical Poincaré residue techinque, the periods of cubic hypersurfaces parametrized by \(Y\) exactly coincide with the periods of the double covers in the Legendre family parametrized by \(Y_L\), as the following lemma explains. Hence, we will use the Galois restriction property to lift the explicit solution \(\varphi(\lambda)\) of the reduced Aomoto-Gelfand system \(E'(2, 4)\) to the entire space \(V\).

We end by remarking on a subtlety in this period formula. In the proof of Theorem 5, we began with the solution \(\varphi(\lambda)\) on \(Y\), defined in a neighborhood in \(Y\) by the hypergeometric function \(\leftindex_2{F}_1 \left( \frac{1}{2}, \frac{1}{2}; 1; \lambda \right)\) and extended to a multivalued function, which we also denote \(\varphi(\lambda)\), on the rest of \(Y\) by analytic continuation. Then we lifted \(\varphi(\lambda)\) to a function \(P(Z) \varphi(\rho(Z))\) on a neighborhood in \(V^o\) where the second factor is constant on \(G\)-orbits, which extends to a multivalued function \(\Pi(Z)\) on all of \(V^o\) by analytic continuation. Each \(G\)-orbit in \(V^o\) intersects \(Y\) at multiple points \(p_i\), and \(P(Z)\) is constant on \(Y\), so by \(\operatorname{SL}_3(\mathbb{C})\)-invariance of \(\varphi(\rho(Z))\) it appears that \(\Pi(Z)\) should take the same value at each \(p_i\) thanks to the analytic lifting lemma, Lemma 5. But the \(p_i\) are transformed into one another by elements of \(G = \operatorname{GL}_3(\mathbb{C})\) with nontrivial determinant, so the transformation properties of the tautological system suggest that \(\Pi(Z)\) should take different values at the \(p_i\). At first glance, this seems contradictory.

Recall, however, that the space of solutions to the Gauss hypergeometric differential equation is two-dimensional, in this case spanned by \(\leftindex_2{F}_1 \left( \frac{1}{2}, \frac{1}{2}; 1; \lambda \right)\) and a modification thereof involving \(\log \lambda\) and a power series in \(\lambda\) whose coefficients involve the digamma function [38]. It turns out that the transformation properties of \(\Pi(Z)\) encoded in the tautological system result from the relationship between the different branches of \(\varphi(\lambda)\); in other words, the transformation properties of \(\Pi(Z)\) under the \(\operatorname{GL}_3(\mathbb{C})\)-action depend on whether one analytically continues \(\Pi\) along \(Y\) or along a \(G\)-orbit.

This discrepancy reflects a tension in the current form of the theory: from the standpoint of tautological systems, the periods should be considered as sheafy multivalued objects, a perspective which is not realized in the reduction procedure of \(E(n, 2n)\). This highlights the pressing need to understand the reduction of \(\mathcal{D}\)-modules to subvarieties in order to adapt the method to more general contexts. We anticipate that invariant theory, and restriction properties in particular, will play a vital role in these questions.

Acknowledgments↩︎

We thank Gerry Schwarz and the anonymous referee for their helpful comments on earlier versions of the manuscript. We also extend deep gratitude to Gerry Schwarz for insightful conversations at the early stages of this work which greatly influenced the direction of our research. K.S. thanks the Shanghai Institute for Mathematics and Interdisciplinary Sciences for their generous hospitality during which part of this work was completed.

References↩︎

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1. An incorrect version of this theorem appeared in an earlier version of this article.↩︎