1 Introduction↩︎

The purpose of this paper is to discuss the duality theory of some differential systems that are naturally attached to group actions on algebraic varieties. More precisely, we are concerned with the so-called tautological systems, which were first considered in [1], discussed from various points of views (especially towards applications to mirror symmetry) in [2]–[5], and then studied thoroughly in our previous paper [6]. There, we were especially interested in Hodge theoretic aspects of tautological systems associated to homogeneous spaces. By the functoriality properties within the category of mixed Hodge modules, it is particularly important to understand the duality theory of such systems. In the present paper, we study the holonomic dual of general tautological systems in detail. Our results can be considered as a generalization of a similar study for the case where the group is an algebraic torus (leading to the well-known GKZ-systems), in that case, it is a result of Walther (see [7]) that under some suitable hypotheses, the dual system is again a GKZ-system defined essentially by the same initial data. One consequence of our findings is that such a direct expression of the holonomic dual holds essentially only if the dimension of the group coincides with that of the variety it acts on (which is obviously true in the toric case). In general, we give a cohomological expression of the holonomic dual (see [prop:DualityTautCM] and 3 below). If tautological systems are constructed functorially (as it is the case for those coming from homogeneous spaces, see [6]), there is a natural morphism between this system and its dual (the direction of which depends on the precise construction by standard functors).

Let us give a more precise overview about the main results of this paper. Throughout, we work over the field \(\mathbb{C}\). To a regular representation \(\rho \colon G \to V\) of a connected reductive algebraic group \(G\), an orbit closure \(\overline{Y} \subseteq V\) and a Lie algebra homomorphism \(\beta \colon \mathfrak g \to \mathbb{C}\), one can associate the Fourier-transformed tautological system \(\hat{\tau}(\rho,\overline{Y}, \beta)\). This is a \(\mathscr D_V\)-module with an explicit cyclic presentation given by the \(\mathscr{D}_V\)-ideal generated by the vanishing ideal of \(\overline{Y} \subseteq V\) and the vector fields on \(V\) induced by the group action of \(G\) with a twist given by \(\beta\). The actual tautological system (called \(\tau(\rho,\overline{Y}, \beta)\) in [6]) is the total Fourier-Laplace transformation of \(\hat{\tau}(\rho,\overline{Y}, \beta)\), that is, a cyclic \(\mathscr{D}_{V^\vee}\)-module on the dual space \(V^\vee\). Since the duality functor and the Fourier-Laplace transformation commute up to sign, it is essentially equivalent to describe \({\mathbb{D}}\tau(\rho,\overline{Y}, \beta)\) and \({\mathbb{D}}\hat{\tau}(\rho,\overline{Y}, \beta)\). For this reason, we will in this paper consider almost exclusively the latter object.

Our main tool to describe the holonomic dual of \(\hat{\tau}(\rho,\overline{Y}, \beta)\) is a Chevalley–Eilenberg type complex, namely, we consider \(\hat{T}^\bullet(\rho, \overline{Y}, \beta) = \mathscr D_V \otimes_{\mathcal{O}_V} \mathcal{O}_{\overline{Y}} \otimes \bigwedge^{-\bullet} \mathfrak g\) with a differential depending on \(\beta\). We comment below in 2 on how our complex relates to similar constructions in Lie algebra theory. Let us remark that in the toric case, our complex reduces to the Euler-Koszul complex studied in the theory of GKZ-systems (see [rmk:EK-complex] below).

One of our main results can be summarized as follows.

A special case where the assumptions of 2 are satisfied and where we have \(\dim\overline{Y}=\dim(G)\) is constructed from so-called linear free divisors. In this case, we use the general duality result to strengthen considerably the main result from [8], namely, we show that in this case \(\hat{\tau}(\rho,\overline{Y},\beta)\) underly a complex mixed Hodge module for all but at most finitely many values of \(\beta\) (see [prop:casesLFD] and [cor:AllButFinteLFD]), and that moreover under a stronger assumption on \(\beta\), the actual tautological system \(\tau(\rho,\overline{Y},\beta)\) associated to a given linear free divisor has an irreducible monodromy representation ([prop:casesLFD], 3.).

In 5.2 we comment about a few more special cases where under additional assumptions we get sharper duality results. We show in particular that if \(\dim(G)=\dim\overline{Y}+1\), then in some cases the dual of \(\hat{\tau}(\rho,\overline{Y},\beta)\) is a again tautological system for \(\rho\) and \(\overline{Y}\) and a possibly different \(\beta\).

One of the main motivations for our work come from a central result of [6] (namely, Theorem 6.14 in loc.cit.), where we studied tautological systems defined by homogeneous spaces: The system \(\tau(\rho, \overline{Y}, \beta)\) (the same statement holds for the system \(\hat{\tau}(\rho, \overline{Y}, \beta)\)) underlies a (in general complex) mixed Hodge module which has a weight filtration of length at most two. Therefore, it is of particular interest to understand the only possible non-trivial weight filtration step. By the functorial construction of \(\hat{\tau}\) (resp. of \(\tau\)) (see again Theorem 6.14, point 2.in loc.cit.), if the weight filtration is non-trivial (i.e., of length two) we have a duality morphism \(\tau(\rho, \overline{Y}, \beta)\rightarrow {\mathbb{D}}\tau(\rho, \overline{Y}, \beta)\), and the non-trivial weight step is the kernel of this morphism. We postpone the study of this duality morphism to a subsequent paper, however, we do make in 5.3 a few comments and conjectures about it. It should be noticed that in the case where the group \(G\) is an algebraic torus, and when considering GKZ-systems defined by an action of a torus, this duality morphism has been computed explicitly in [9] based on [10]. This computation is a key step in establishing a mirror symmetry statement for nef complete interse ctions in toric varieties using non-affine Landau-Ginzburg models (see [9]).

Acknowledgements: We would like to thank Thomas Reichelt and Uli Walther for many fruitful discussions about the topic of this paper and Christian Lehn and Patrick Graf for answering our questions about dualizing modules.

2 Duality for Lie algebroids↩︎

In this section, we exhibit a general duality result in the realm of Lie algebroids, which we will in later sections apply to our specific situation.

2.1 Chevalley–Eilenberg complexes for Lie algebroids↩︎

We recall for the reader’s convenience basic facts about Lie algebroids and their Chevalley–Eilenberg complexes and fix notations. Most of them are easily found in the literature, see [11], [12]. We also prove a certain self-duality result on the level of the Chevalley–Eilenberg complex for Lie algebroids, for related statements, see e.g.[13].

We recall that a Lie algebroid \((\mathcal{E}, [\cdot, \cdot], Z)\) on a smooth complex variety \(X\) is an \(\mathcal{O}_X\)-module \(\mathcal{E}\) equipped with a Lie bracket \([\cdot, \cdot] \colon \mathcal{E} \times \mathcal{E} \to \mathcal{E}\) (assumed to be \(\mathbb{C}\)-bilinear, alternating and satisfying the Jacobi identity), making \(\mathcal{E}\) a sheaf of Lie algebras over \(\mathbb{C}\), and together with a map \(Z \colon \mathcal{E} \to \Theta_X\), called the anchor map of \(\mathcal{E}\), which is both a morphism of \(\mathcal{O}_X\)-modules and a morphism of sheaves of Lie algebras over \(\mathbb{C}\). We will mostly work with Lie algebroids \((\mathcal{E}, [\cdot, \cdot], Z)\) such that the \(\mathcal{O}_X\)-module \(\mathcal{E}\) is locally free of finite rank. We often speak of a Lie algebroid \(\mathcal{E}\), with the data of the Lie bracket and the anchor map understood implicitly.

Given a Lie algebroid \(\mathcal{E}\), there is a universal enveloping algebra \(\mathcal{U}(\mathcal{E})\). We will consider (left or right) \(\mathcal{U}(\mathcal{E})\)-modules which will always be assumed to be \(\mathcal{O}_X\)-quasi-coherent. If \(\mathcal{E}\) is locally free of finite type as \(\mathcal{O}_X\)-module, a PBW-theorem holds for \(\mathcal{U}(\mathcal{E})\) (see, e.g., [11]), i.e., there is canonical filtration on \(\mathcal{U}(\mathcal{E})\) such that its graded algebra is isomorphic to the symmetric algebra (over \(\mathcal{O}_X\)) of \(\mathcal{E}\). In particular, by standard arguments in homological algebra (see, e.g., [14] or [15]), \(\mathcal{U}(\mathcal{E})\) has finite global dimension. Henceforth, when speaking of the derived category of \(\mathcal{U}(\mathcal{E})\)-modules, we always mean the correponding bounded derived category.

For a Lie algebroid \(\mathcal{E}\) that is locally free of rank \(m\) as an \(\mathcal{O}_X\)-module, the line bundle \(\omega_{\mathcal{E}} := \bigwedge_{\mathcal{O}_X}^{m} \mathcal{E}^\vee\) has a right \(\mathcal{U}(\mathcal{E})\)-module structure, see, e.g.[14]. The tensor product over \(\mathcal{O}_X\) of two left (resp.a left and a right) \(\mathcal{U}(\mathcal{E})\)-modules is again a left (resp.right) \(\mathcal{U}(\mathcal{E})\)-module as in the theory of \(\mathscr D\)-modules, see e.g.[16] for the case of \(\mathscr{D}\)-modules or [14] or [13] for more general Lie algebroids. Similarly, \(\mathop{\mathrm{\mathscr{H}\kern -3pt\textit{om}\kern 1pt}}_{\mathcal{O}_X}(\mathcal{M}, \mathcal{N})\) of two left (or two right) \(\mathcal{U}(\mathcal{E})\)-modules \(\mathcal{M}\), \(\mathcal{N}\) is a left \(\mathcal{U}(\mathcal{E})\)-module.

In particular, if \(\mathcal{M}\) is a left \(\mathcal{U}(\mathcal{E})\)-module, then \(\omega_\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{M}\) is a right \(\mathcal{U}(\mathcal{E})\)-module. Conversely, if \(\mathcal{M}'\) is a right \(\mathcal{U}(\mathcal{E})\)-module, then \(\mathcal{M}' \otimes_{\mathcal{O}_X} \omega_{\mathcal{E}}^\vee\) is a left \(\mathcal{U}(\mathcal{E})\)-module.

For any Lie algebroid \(\mathcal{E}\) which is locally free of finite rank \(m\) over \(\mathcal{O}_X\), with universal enveloping algebra \(\mathcal{U} := \mathcal{U}(\mathcal{E})\), we consider the duality functor \[\mathbb{D}_{\mathcal{U}} := R\!\mathop{\mathrm{\mathscr{H}\kern -3pt\textit{om}\kern 1pt}}_{\mathcal{U}}(\cdot,\mathcal{U}) \otimes_{\mathcal{O}_X} \omega_{\mathcal{E}}^\vee [m]\] in the derived category of left \(\mathcal{U}\)-modules. While in this section, we deal with general Lie algebroids, we will later apply the discussion here to one of the following examples:

Notice that, already from these two basic examples, modules over universal enveloping algebras of Lie algebroids can be seen as a uniformizing framework including as special cases both \(\mathfrak g\)-modules and \(\mathscr D_X\)-modules.

We next consider the Chevalley–Eilenberg complex of a Lie algebroid.

Then \(\mathcal{S}_{\mathcal{R}|\mathcal{U}}^\bullet(\mathcal{N})\) is a complex of left \(\mathcal{R}\)-modules (one checks that \(\delta^{-\ell} \circ \delta^{-\ell+1} = 0\)). One can prove that \(\mathcal{S}_{\mathcal{R}|\mathcal{U}}\) is an exact functor from the abelian category of \((\mathcal{R},\mathcal{U})\)-bimodules to the abelian category of complexes of left \(\mathcal{R}\)-modules.

In the case of [ex:AModAsLieAlgebroid], when we consider a Lie algebroid of the form \(\mathcal{E} = \mathcal{O}_X \otimes \mathfrak g\) for a \(G\)-action on \(X\), we have the following comparision result relating the above complex with the corresponding object in the theory of Lie algebras.

Notation: Throughout, let \(X\) be a connected smooth complex variety. We will for brevity of notation often write \(\mathcal{O}\) instead of \(\mathcal{O}_X\).

The following basic fact is an analogue of [16]:

This is proven, e.g., in [11] (see also [12]).

Notice that the statement of the previous lemma can also directly be proven by considering the filtration on \(\mathcal{N} = \mathcal{U} \otimes_{\mathcal{O}_X} \mathcal{M}\) induced by the PBW-filtration on \(\mathcal{U}\) and passing to the graded objects, see [6]. Then we obtain [lem:resolutionOfO] as a special case when \(\mathcal{M}=\mathcal{O}_X\).

We next observe see a property that can be seen as a certain self-duality of the Chevalley–Eilenberg complex:

2.2 Duality of \(\mathcal{U}(\mathcal{E})\)-modules↩︎

In the following, we consider two Lie algebroids \(\mathcal{E}_1\) and \(\mathcal{E}_2\) on \(X\) locally free over \(\mathcal{O}\) of finite ranks \(m_1\) and \(m_2\), respectively. Let \(\mathcal{U}_1\) and \(\mathcal{U}_2\) denote their universal enveloping algebras. Recall that \(\omega_i := \bigwedge_{\mathcal{O}}^{m_i} \mathcal{E}_i^\vee\) has a right \(\mathcal{U}_i\)-module structure (given by the negated Lie derivative).

We consider the following double side-changing operation on bimodules: If \(\mathcal{M}\) is a \((\mathcal{U}_2,\mathcal{U}_1)\)-bimodule, then \(\omega_2 \otimes_{\mathcal{O}} \mathcal{M} \otimes_{\mathcal{O}} \omega_1^\vee\) is a \((\mathcal{U}_1,\mathcal{U}_2)\)-bimodule. Here, note that the first tensor product uses the \(\mathcal{O}\)-module structure via \(\mathcal{O}\to \mathcal{U}_2\) and the left module structure of \(\mathcal{M}\) and the second tensor product uses the \(\mathcal{O}\)-module structure via \(\mathcal{O}\to \mathcal{U}_1\) and the right module structure of \(\mathcal{M}\). In general, when working with \((\mathcal{U}_1,\mathcal{U}_2)\)-bimodules, we always write tensor products in an order that makes implicitly clear which \(\mathcal{O}\)-module structure is being used.

It is straightforward to check the following basic property:

If \(\mathcal{N}\) is a \((\mathcal{U}_1, \mathcal{U}_2)\)-bimodule, then \(\mathop{\mathrm{\mathscr{H}\kern -3pt\textit{om}\kern 1pt}}_{\mathcal{U}_1}(\mathcal{N}, \mathcal{U}_1)\) naturally inherits a \((\mathcal{U}_2,\mathcal{U}_1)\)-bimodule structure from the right \(\mathcal{U}_2\)-module structure on \(\mathcal{N}\) and the right \(\mathcal{U}_1\)-module structure on \(\mathcal{U}_1\). We obtain from [lem:dualityGeneralized] the following duality property:

Assume now that there is a homomorphism of Lie algebroids \(\mathcal{E}_2 \to \mathcal{E}_1\). This induces a \(\mathbb{C}\)-algebra homomorphism \(\mathcal{U}_2 \to \mathcal{U}_1\) between their universal enveloping algebras. In particular, every (left or right) \(\mathcal{U}_1\)-module may be viewed as a (left or right) \(\mathcal{U}_2\)-module via \(\mathcal{U}_2 \to \mathcal{U}_1\). On the other hand, for a left \(\mathcal{U}_2\)-module \(\mathcal{M}\), we may consider \(\mathcal{U}_1 \otimes_{\mathcal{U}_2}^\mathbb{L} \mathcal{M}\) in the derived category of \(\mathcal{U}_1\)-modules. This object is represented by the complex \(\mathcal{S}^\bullet_{\mathcal{U}_1|\mathcal{U}_2}(\mathcal{U}_1 \otimes_{\mathcal{O}} \mathcal{M})\), because \[\mathcal{S}^\bullet_{\mathcal{U}_1|\mathcal{U}_2}(\mathcal{U}_1 \otimes_\mathcal{O}\mathcal{M}) \cong (\mathcal{U}_1 \otimes_\mathcal{O}^\mathbb{L} \mathcal{M}) \otimes_{\mathcal{U}_2}^\mathbb{L} \mathcal{O} \cong (\mathcal{U}_1 \otimes_\mathcal{O}^\mathbb{L} \mathcal{O}) \otimes_{\mathcal{U}_2}^\mathbb{L} \mathcal{M} \cong \mathcal{U}_1 \otimes_{\mathcal{U}_2}^\mathbb{L} \mathcal{M},\] where the first isomorphism is due to [lem:ComplexSRepresents] and the second isomorphism is due to [lem:HTTanalogue].

In this context, the right \(\mathcal{U}_1\)-module \(\omega_1\) can also be viewed as a right \(\mathcal{U}_2\)-module via \(\mathcal{U}_2 \to \mathcal{U}_1\), which together with the right \(\mathcal{U}_2\)-module structure on \(\omega_2\) makes \[\omega_{1|2} := \mathop{\mathrm{\mathscr{H}\kern -3pt\textit{om}\kern 1pt}}_\mathcal{O}(\omega_1,\omega_2)\] a left \(\mathcal{U}_2\)-module. This is the relative dualizing module in [14].

Through \(\mathcal{U}_2 \to \mathcal{U}_1\), the algebra \(\mathcal{U}_1\) is in particular a \((\mathcal{U}_2,\mathcal{U}_1)\)-bimodule or a \((\mathcal{U}_1,\mathcal{U}_2)\)-bimodule. In fact, these two bimodule structures are related to each other under double side-changing via \(\omega_{1|2}\) as follows:

More generally, we get:

The following is a consequence of the (self-)duality result for Chevalley-Eilenberg complexes (i.e. [lem:dualityGeneralized]). We would like to point out that the duality result from [14] expresses the duality functor \(\mathbb{D}_{\mathcal{U}_1}(-)\) using \(\mathbb{D}_{\mathcal{U}_2}(-)\) (and is therefore rather a extension of scalar type property) whereas our result expresses \(\mathbb{D}_{\mathcal{U}_1}(-)\) using \(R\!\mathop{\mathrm{\mathscr{H}\kern -3pt\textit{om}\kern 1pt}}_{\mathcal{O}_X}(-,\mathcal{O}_X)\).

Taking \(\mathcal{U}_1 = \mathcal{U}_2\), this leads to the following:

This corollary is a generalization of [12], in the sense that in loc.cit.it is assumed that \(\mathcal{M}\) is \(\mathcal{O}_X\)-locally free, whereas we only need \(\mathcal{O}_X\)-coherence. Note that for \(\mathcal{U} = \mathscr D_X\), assuming \(\mathcal{M}\) to be \(\mathcal{O}_X\)-coherent obviously implies that it is an integrable connection, in which case \(\mathbb{D}(\mathcal{M}) \cong \mathop{\mathrm{\mathscr{H}\kern -3pt\textit{om}\kern 1pt}}_{\mathcal{O}_X}(\mathcal{M}, \mathcal{O}_X)\) (see, e.g., [16]). However, for other Lie algebroids, there are cases of \(\mathcal{O}\)-coherent left \(\mathcal{U}\)-modules \(\mathcal{M}\) which are not \(\mathcal{O}\)-locally free, and, consequently, for which \(\mathbb{D}_{\mathcal{U}}(\mathcal{M})\) is not a single \(\mathcal{U}\)-module in cohomological degree \(0\).

3 Equivariant aspects↩︎

In this section, we study group actions on smooth varieties which determines a morphism of Lie algebroids as in [ex:AModAsLieAlgebroid]. The results from 2 can then be made more explicit. Along the way, we consider equivariance properties of the complexes involved and deduce a holonomicity criterion.

3.1 Holonomicity↩︎

Throughout, we consider the action of a connected linear algebraic group \(G\) on a smooth connected variety \(X\). We introduce a complex \(\mathcal{C}^\bullet(\mathcal{M},\beta)\) of \(\mathscr D_X\)-modules associated to a \(G\)-equivariant quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{M}\) and a Lie algebra homomorphism \(\beta \colon \mathfrak g \to \mathbb{C}\), where \(\mathfrak g\) denotes the Lie algebra of \(G\).

For this, we consider the algebra \(\mathcal{A}_X = \mathcal{O}_X \otimes U(\mathfrak g)\) as in [ex:AModAsLieAlgebroid] and in [lem:ComparisonLieAlgHom]. We view the \(G\)-equivariant quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{M}\) as a left \(\mathcal{A}_X\)-module through the \(\mathfrak g\)-action obtained by differentiating. Explicitly, the \(G\)-equivariance of \(\mathcal{M}\) is given by an isomorphism \(\varphi \colon \mathop{\mathrm{\mathrm{act}}}^* \mathcal{M}\to \mathop{\mathrm{\mathrm{pr}}}_2^* \mathcal{M}\), where \(\mathop{\mathrm{\mathrm{act}}}\colon G \times X \to X\) is given by \((g,y) \mapsto g \cdot y\). Any \(\xi \in \mathfrak g\) corresponds to a morphism \(i_\xi \colon \mathop{\mathrm{\mathrm{Spec}\,}}\mathbb{C}[\varepsilon]/\varepsilon^2 \to G\). Then pulling back \(\varphi\) under \(i_\xi \times \mathop{\mathrm{\mathrm{id}}}_X \colon \mathop{\mathrm{\mathrm{Spec}\,}}\mathbb{C}[\varepsilon]/\varepsilon^2 \times X \to G \times X\) yields an isomorphism \[(i_\xi \times \mathop{\mathrm{\mathrm{id}}}_X)^*\varphi \colon \mathbb{C}[\varepsilon]/\varepsilon^2 \otimes \mathcal{M}\xrightarrow{\cong} \mathbb{C}[\varepsilon]/\varepsilon^2 \otimes \mathcal{M}.\] The image of \(1 \otimes m\) under this isomorphism is of the form \(1 \otimes m - \varepsilon \otimes m'\). Then \(\xi \cdot m := m'\) defines the action of \(\mathfrak g\) on \(\mathcal{M}\) which gives the \(\mathcal{A}_X\)-module structure on \(\mathcal{M}\).

In the following, let \(G\) be a connected linear algebraic group with Lie group \(\mathfrak g\), let \(X\) be a connected smooth \(G\)-variety and let \(\mathcal{M}\) be a \(G\)-equivariant quasi-coherent \(\mathcal{O}_X\)-module. Let \(\beta \colon \mathfrak g \to \mathbb{C}\) be a Lie algebra homomorphism.

Then \(\mathcal{C}^\bullet(\mathcal{M},\beta)\) represents \(\mathscr D_X \otimes_{\mathcal{A}_X}^\mathbb{L} \mathcal{M}\{\beta\}\).

3.2 Duality theory↩︎

Our aim is to use [prop:dualOfScalarExtension] to describe the dual of \(\mathcal{C}^\bullet(\mathcal{M}, \beta)\). For this, we need a little preparatory work to describe the relative dualizing module \(\omega_{\mathscr D_V | \mathcal{A}_V}\) associated to the homomorphism of Lie algebroids \(\mathcal{O}_X \otimes \mathfrak g \to \Theta_X\).

The right \(\mathcal{U}(\mathcal{E})\)-module \(\omega_\mathcal{E}\) for a locally free Lie algebroid \(\mathcal{E}\) of finite rank is in the case \(\mathcal{E}= \Theta_X\), \(\mathcal{U}(\mathcal{E})=\mathscr D_X\) (for some smooth variety \(X\)) the canonical sheaf \(\omega_X\) with a right \(\mathscr D_V\)-action via the Lie derivative. In the case \(\mathcal{E}= \mathcal{O}_X \otimes \mathfrak g\), \(\mathcal{U}(\mathcal{E}) = \mathcal{A}_X\), we denote \(\omega_{\mathcal{O}_X \otimes \mathfrak g}\) by \(\alpha_X\), a right \(\mathcal{A}_X\)-module whose underlying \(\mathcal{O}_X\)-module is isomorphic to \(\mathcal{O}_X\).

4 Tautological systems↩︎

To a regular representation \(\rho \colon G \to V\) of a connected linear algebraic group \(G\), the closure \(\overline{Y} \subseteq V\) of a \(G\)-orbit \(Y\) and a Lie algebra homomorphism \(\beta \colon \mathfrak g \to \mathbb{C}\), one associates the following cyclic left \(\mathscr D_V\)-module, the Fourier-transformed tautological system: \[\hat{\tau}(\rho,\overline{Y},\beta):= \mathscr D_V/(\mathscr D_V \mathcal{I}_{\overline{Y}} + \mathscr D_V (Z_V(\xi)-\beta'(\xi) \mid \xi \in \mathfrak g)),\] where \(\mathcal{I}_{\overline{Y}} \subseteq \mathcal{O}_{\overline{Y}}\) is the ideal sheaf of \(\overline{Y} \subseteq V\) and \(\beta':=\mathop{\mathrm{trace}}\circ \mathop{\mathrm{d\!}}\rho-\beta \colon \mathfrak g \to \mathbb{C}\).

This notation originates from the fact that the operator \(Z_V(\xi)-\beta'(\xi)\) is the negated Fourier transform on \(V\) of the operator \(Z_V(\xi)-\beta(\xi)\); those operators show up in the definition of tautological systems \(\tau(\rho,\overline{Y}, \beta) = \mathop{\mathrm{\mathrm{FL}}}(\hat{\tau}(\rho, \overline{Y}, \beta))\). Notice that because of \(\mathop{\mathrm{\mathrm{FL}}}\circ \mathbb{D}_{\mathscr D_V} = -\mathbb{D}_{\mathscr D_V} \circ \mathop{\mathrm{\mathrm{FL}}}\), studying the dual of a tautological system or of its Fourier-transform are equivalent problems, and we choose to wwork with the Fourier-transformed tautological system \(\hat{\tau}(\rho, \overline{Y}, \beta)\).

Note that, using the \(G\)-action on \(V\), we have \[\begin{align} \hat{\tau}(\rho, \overline{Y}, \beta) &= \mathscr D_V \otimes_{\mathcal{A}_V} \mathcal{A}_V/(\mathcal{A}_V \mathcal{I}_{\overline{Y}} + \mathcal{A}_V (\xi-\beta'(\xi) \mid \xi \in \mathfrak g')) \\ &\cong \mathscr D_V \otimes_{\mathcal{A}_V} ((\mathcal{A}_V/\mathcal{A}_V \mathcal{I}_{\overline{Y}})\{\beta'\} \otimes_{\mathcal{A}_V} \mathcal{O}_V)\\ &\cong \mathscr D_V \otimes_{\mathcal{A}_V} ((\mathcal{A}_V \otimes_{\mathcal{O}_V} \mathcal{O}_{\overline{Y}}\{\beta'\})\otimes_{\mathcal{A}_V} \mathcal{O}_V) \\ &\cong \mathscr D_V \otimes_{\mathcal{A}_V} \mathcal{O}_{\overline{Y}}\{\beta'\}, \end{align}\] where the last isomorphism is due to [lem:HTTanalogue].

By [lem:ComplexSRepresents], in the derived category \(D_{qc}^b(\mathscr D_V)\) of left \(\mathscr D_V\)-modules, we have \[\hat{T}(\rho, \overline{Y}, \beta) \cong (\mathscr D_V \otimes_{\mathcal{O}_V} \mathcal{O}_{\overline{Y}}\{\beta'\}) \otimes_{\mathcal{A}_V}^\mathbb{L} \mathcal{O}_V \cong \mathscr D_V \otimes_\mathcal{A}^\mathbb{L} \mathcal{O}_{\overline{Y}}\{\beta'\}\] where the last isomorphism is due to [lem:HTTanalogue]. In particular: \[\hat{\tau}(\rho,\overline{Y}, \beta) = H^0 \hat{T}(\rho, \overline{Y}, \beta),\] which is also easily checked from the definition of \(\mathcal{S}_{\cdot|\cdot}^\bullet(\cdot)\) itself.

From [cor:holonomy] and [cor:CohomAmplitude], we immediately obtain:

In the following, we will consider the more restrictive case where \(\overline{Y}\) is supposed to be Gorenstein with trivial canonical bundle. More precisely, we assume that there is a Lie algebra homomorphism \(\gamma \colon \mathfrak g \to \mathbb{C}\) such that \(\omega_{\overline{Y}} \cong \mathcal{O}_{\overline{Y}}\{-\gamma\}\) as left \(\mathcal{A}_V\)-modules (where \(\omega_{\overline{Y}}\) carries a left \(\mathcal{A}_V\)-structure since it is an \(G\)-equivariant \(\mathcal{O}_{\overline{Y}}\)-module, see [conv:OmegaM]).

We now consider the following situation: For this, let \(\rho_0 \colon G_0 \to \mathop{\mathrm{\mathrm{GL}}}(V)\) be a representation of a group \(G_0\). We extend this to a representation \(\rho \colon G \to \mathop{\mathrm{\mathrm{GL}}}(V)\) for \(G := \mathbb{C}^* \times G_0\) by letting \(\mathbb{C}^*\) act by simple scaling on \(V\). Let \(\overline{Y} \subseteq V\) be a \(G\)-orbit closure, which is then the affine cone over a projective \(G_0\)-variety \(Z \subseteq \mathbb{P}V\). In this setup, if \(\overline{Y}\) is Gorenstein, then its canonical bundle is trivial (since \(\mathop{\mathrm{Pic}}(\overline{Y}) = 0\)). Then by considering a trivializing section of \(\omega_{\overline{Y}}\), it follows that \(G\) must act via a group character \(\chi:G\rightarrow {\mathbb{C}}^*\), and then we have \(\omega_{\overline{Y}} \cong \mathcal{O}_{\overline{Y}}\{-\gamma\}\), where \(\gamma\colon \mathfrak{g}\to{\mathbb{C}}\) is the derivative of \(\chi^{-1}\).

5 Applications and Examples↩︎

In the setup of the previous section, we can give more precise information on the dual of a tautological system under some assumptions on the dimension of the group \(G\) compared to the dimension of the (closure of the) \(G\)-orbit \(\overline{Y}\) used in the definition of the tautological system. We list below a few cases of interest.

5.1 Case \(\dim(G)=\dim(\overline{Y})\)↩︎

In this case, we obtain the most satisfying result as a direct consequence of 3:

We observe that the class of GKZ-systems given by a matrix of full row-rank fall into this case where the dimension of the group and the orbit agree. [cor:DualDimEqual] generalizes the duality result for GKZ-systems [7].

Application: As a specific example of the construction described above, we consider a situation arising when studying so-called linear free divisors. We refer specifically to the setup of [8]. Namely, consider a reductive algebraic group acting prehomogeneously on a complex vector space \(V_0\) (and denote by \(x_1, \ldots, x_n\) a coordinate system relative to a chosen basis of \(V_0\)), meaning that there is a dense open orbit in \(V_0\). Additionally, we require that the stabilizers for all points of this open orbit are finite groups. Let \(D\) be the complement of the open orbit, and assume that \(D\subseteq V\) is a divisor. The group we started with can then be characterized as \(G_D:=\{g\in\mathop{\mathrm{\mathrm{GL}}}(V)\,|\,gD=D\}\). We also have \(\dim(G_D)=\dim(V_0)=n\). Moreover, let \(\xi_1,\ldots,\xi_n\in \mathfrak{g}_D\) be a basis of the Lie algebra of \(G_D\), then let \(A=(a_{ij})\in\mathrm{Mat}(n\times n, {\mathbb{C}}[V]_1\)) such that \(Z_V(\xi_j)=\sum_{i=1}^n a_{ij}\partial_{x_i}\) (where \(Z_V\) is the notation for the anchor map used in [ex:AModAsLieAlgebroid]). Then we require that the determinant \(f:=\det(A)\in \mathcal{O}_V\) is reduced (it is then automatically a reduced equation of \(D\), and necessarily we have that \(f\in{\mathbb{C}}[V]_n\)). Under these conditions, we have that the module of vector fields \(\Theta_V(-\log\,D)\) logarithmic along \(D\) is free, and \(D\) is called a linear free divisor (it is a free divisor in the sense of K. Saito, see [23], and it is called linear free since the polynomials \(a_{ij}\) are linear forms on \(V\)). We will later further restrict to the class of so-called strongly Koszul (SK) free divisors, which in the current situation can simply be characterized by saying that the action of \(G_D\) on \(V\) has finitely many orbits.

We briefly recall some of the main constructions in [8]. First notice that there is a subgroup \(G_0 \subset G_D\) (with \(\dim(G_0)=n-1\)) consisting of all linear transformations that stabilizes all fibres of \(f\) (which was called \(A_D\) in loc.cit.). Then we put \(G:={\mathbb{C}}^*\times G_0\), \(V={\mathbb{C}}\times V_0\) and we consider the extended action \[\begin{array}{rcl} \rho: G & \longrightarrow & \mathop{\mathrm{\mathrm{Aut}}}(V) \\ \\ (t,g) & \longmapsto & (v\mapsto t\cdot \rho_0(g)(v)), \end{array}\] where \(\rho_0\) is the restriction of the original prehomogeneous action to \(G_0\subset G_D\). Let \(Y_0:=f^{-1}(t)\) for \(t\in{\mathbb{C}}^*\), then \(Y_0\) is a (closed) \(G_0\)-orbit. Let \(Y:=C(Y_0)\) be its cone in \({\mathbb{C}}^*\times V_0\), i.e. \(Y=\rho(G)(1,p)\) for any point \(p\in Y_0\). In particular, we have an isomorphism \(\iota:{\mathbb{C}}^*\times Y_0\rightarrow Y\), \((t,y)\mapsto (t,t\cdot y)\). Then \(Y\) is a (usually non-closed) \(G\)-orbit, and we let \(\overline{Y}\subset V\) be its closure. Clearly, the boundary \(\partial Y\) is contained in \(\{0\}\times V\). The main results of [8] then concerns the tautological system \(\tau(\rho,\overline{Y},\beta)\). For simplicity, we will assume that \(G_0\) is semi-simple, which then implies that \(\mathop{\mathrm{trace}}\circ \mathop{\mathrm{ad}}= 0\) and that \(\beta_{|\mathfrak{g}_0}=0\), where \(\mathfrak{g}={\mathbb{C}}\mathbf{e} \oplus \mathfrak{g}_0\). We cite the following result (notice that the various shifts and sign differences occuring here are due to the change of convention for the Euler field \(Z_V(\mathbf{e})\) and for the definition of the module \(\mathcal{O}_{{\mathbb{C}}^*}^{\beta(\mathbf{e})}\))

From the basic functorial properties of the holonomic duality functor, we obtain the following consequence.

Combining this last statement with our duality result [cor:DualDimEqual] then yields the following.

Note that the intersection of \(n \cdot (1+\mathop{\mathrm{\mathrm{roots}}}(b_D)) + \mathbb{Z}_{>0}\) and \(n \cdot (1+\mathop{\mathrm{\mathrm{roots}}}(b_D)) + \mathbb{Z}_{\leq 0}\) is a finite set. Therefore, the above in particular shows:

In some special cases, where the roots of \(b_D\) are known, we get an even sharper result. In particular, we know by [24] that for the linear free divisors which are discriminants in representation spaces for quivers of types \(A_n\), \(E_6\) and \(D_m\) if \(3 \nmid m-1\) that no two roots of \(b_D\) differ by a multiple of \(1/n\), so that the intersection of \(n \cdot (1+\mathop{\mathrm{\mathrm{roots}}}(b_D)) + \mathbb{Z}_{>0}\) and \(n \cdot (1+\mathop{\mathrm{\mathrm{roots}}}(b_D)) + \mathbb{Z}_{\leq 0}\) is necessarily empty, so that \(\hat{\tau}(\rho, \overline{Y}, \beta)\) underlies an object in \(\mathrm{MHM}(V,{\mathbb{C}})\) for all \(\beta(\mathbf{e})\) in these cases.

5.2 Case \(\dim(G)=\dim(\overline{Y})+1\)↩︎

We consider again the case of a representation \(\rho_0 \colon G_0 \to V\) which we extend with the scaling action on \(V\) to a representation \(\rho \colon G \to V\) for \(G = \mathbb{C}^* \times G_0\). Let \(\overline{Y}\) be a \(G\)-orbit closure (this is in particular the affine cone over a projective \(G_0\)-variety \(Z \subseteq \mathbb{P}V\)). We then have the following general lemma (which does not depend on assumptions on \(\dim(G)\) and \(\dim(\overline{Y})\)).

We remark that we choose to consider \(\dim V - Z_V(\mathbf{e})\) for \(\hat{\tau}(\rho_0,\overline{Y}, \beta_0)\) because, after Fourier-transform, this becomes the Euler vector field \(E\) on \(V^\vee\), meaning that \(b_{\beta_0}(E)\) lies in the ideal of \(\tau(\rho_0, \overline{Y}, \beta_0)\).

5.3 Homogeneous spaces↩︎

Consider again the case of the affine cone \(\overline{Y} \subseteq V\) over a projective homogeneous space \(X:=G_0/P\) as in [ex:HomSpace]. Here, \(V := \Gamma(X,\mathscr{L})^\vee\) for a \(G\)-equivariant very ample line bundle on \(X\) for \(G = \mathbb{C}^* \times G_0\). Assume \(G_0\) is semisimple, so that every Lie algebra homomorphism \(\beta\colon \mathfrak g = \mathfrak g_0 \oplus \mathbb{C}\mathbf{e} \to \mathbb{C}\) is given by \({\beta_{| {\mathfrak g_0}}} = 0\) and \(\beta(\mathbf{e}) \in \mathbb{C}\).

Let \(\beta(\mathbf{e}) = \ell \in \mathbb{Z}_{>0}\) and assume \(\mathscr{L}^{\otimes \ell} = \omega_{G_0/P}^\vee\) (which is then the only case in which \(\tau(\rho, \overline{Y}, \beta)\) is non-zero). In [6], it was shown that \[\hat{\tau}(\rho, \overline{Y}, \beta) = H^0\iota_\dagger \mathcal{O}_{\overline{Y} \setminus 0},\] where \(\iota \colon \overline{Y} \setminus 0 \hookrightarrow V\). Then \[\mathbb{D}\hat{\tau}(\rho, \overline{Y}, \beta) = H^0\iota_+ \mathcal{O}_{\overline{Y}}\] and by 2 in [ex:HomSpace] this is \(H^{n-m}\hat{T}(\rho, \overline{Y}, 0)\), where \(n := \dim \overline{Y}\), \(m:= \dim G\).

In particular, we have a natural morphism \[\label{eq:morphismToDual} \hat{\tau}(\rho, \overline{Y}, \beta) = H^0 \iota_\dagger \mathcal{O}_{\overline{Y} \setminus 0} \to H^0 \iota_+ \mathcal{O}_{\overline{Y} \setminus 0} = \mathbb{D} \hat{\tau}(\rho, \overline{Y}, \beta) = H^{n-m} \hat{T}(\rho, \overline{Y}, 0)\tag{4}\] whose image is \(\iota_{\dagger +}\mathcal{O}_{\overline{Y}\backslash 0}\), i.e, the intersection cohomology \(\mathscr D\)-module of the singular variety \(\overline{Y}\).

In order to given an explicit description of \(\iota_{\dagger +}\mathcal{O}_{\overline{Y}\backslash 0}\) it would therefore be of interest to describe the morphism \(\hat{\tau}(\rho, \overline{Y}, \beta) \to H^{n-m}\hat{T}(\rho, \overline{Y}, 0)\) explicitly. Recall that the latter is the cohomology of the complex \(\mathscr D_V \otimes_{\mathcal{O}_V} \mathcal{O}_{\overline{Y}} \otimes_{\mathcal{O}_V} \bigwedge_{\mathcal{O}_V}^{-\bullet} (\mathcal{O}_V \otimes \mathfrak g)\) in cohomological degree \(m-n = \dim G_0 - \dim X = \dim P\). It is natural to suspect that the parabolic subgroup \(P\) and its Lie algebra \(\mathfrak p\) lead through \(\dim \bigwedge^{m-n} \mathfrak p = 1\) to a class of the \((n-m)\)-th cohomology, to which the generator of the cyclic \(\mathscr D_V\)-module \(\hat{\tau}(\rho, \overline{Y}, \beta)\) maps under 4 . More specifically, the locally free Lie algebroid \(\mathcal{E} := \ker(\mathcal{O}_Y \otimes \mathfrak g \twoheadrightarrow \Theta_Y)\) of rank \(m-n\) on \(Y := \overline{Y} \setminus \{0\}\) has determinant \(\bigwedge_{\mathcal{O}_Y}^{m-n} \mathcal{E} \cong \bigwedge_{\mathcal{O}_Y}^m (\mathcal{O}_Y \otimes \mathfrak g) \otimes_{\mathcal{O}_Y} \omega_Y^\vee \cong \mathcal{O}_Y\) by the Gorenstein property of \(\overline{Y}\) as discussed in [ex:HomSpace]. This leads to a distinguished global section of \(\bigwedge^{m-n}_{\mathcal{O}_Y} \mathcal{E} \subseteq \bigwedge^{m-n}_{\mathcal{O}_Y} (\mathcal{O}_Y \otimes \mathfrak g)\) and therefore of \(\mathcal{O}_{\overline{Y}} \otimes \bigwedge^{m-n}_{\mathcal{O}_V} (\mathcal{O}_V \otimes \mathfrak g)\) which defines a class in \(H^{n-m} \hat{T}(\rho, \overline{Y}, 0)\). We expect that the generator of \(\hat{\tau}(\rho, \overline{Y}, \beta)\) is mapped to this global section by the morphism 4 . We postpone the details of this construction to a subsequent paper.

Paul Görlach
Otto-von-Guericke-Universität Magdeburg
Fakultät für Mathematik
Institut für Algebra und Geometrie
Universitätsplatz 2
39106 Magdeburg
Germany
paul.goerlach@ovgu.de
Christian Sevenheck
Fakultät für Mathematik
Technische Universität Chemnitz
09107 Chemnitz
Germany
christian.sevenheck@mathematik.tu-chemnitz.de

References↩︎