Hypergeometric Hodge modules.

*(English)*Zbl 1453.14058The paper under review is the third stage of an outstanding project on the Hodge theoretic properties of GKZ systems and their application to mirror symmetry for nef complete intersections in toric varieties, previously appeared in [T. Reichelt and C. Sevenheck, J. Algebr. Geom. 24, No. 2, 201–281 (2015; Zbl 1349.14139); Ann. Sci. Éc. Norm. Supér. (4) 50, No. 3, 665–753 (2017; Zbl 1395.14033)].

Let \(A\) be a \(d\times n\) integer matrix with \(\mathbb{Z}A=\mathbb{Z}^d\) and \(\beta\in\mathbb{C}^d\) be a parameter vector. Take \(\mathbb{C}=\operatorname{Spec}\mathbb{C}[\lambda_1,\ldots,\lambda_n]\) and let \(\mathbb{L}\) be the kernel of \(A\), thought of as a map from \(\mathbb{Z}^n\) to \(\mathbb{Z}^d\). With this notation, define the GKZ system \(\mathcal{M}_A^\beta\) as the cyclic \(\mathcal{D}_{\mathbb{C}^n}\)-module given by \(\mathcal{D}_{\mathbb{C}^n}/\mathcal{I}_A\), where \(\mathcal{I}_A\) is the left ideal generated by the Euler operators \(\sum_j a_{ij}\lambda_j\partial_{\lambda_j}-\beta_i\) for \(i=1,\ldots,d\), and the toric operators \(\prod_{i:l_i>0}\partial_{\lambda_i}^{l_i}-\prod_{i:l_i<0}\partial_{\lambda_i}^{-l_i}\) for every \(l\in\mathbb{L}\).

Such a \(\mathcal{D}\)-module inherits the order filtration from \(\mathcal{D}_{\mathbb{C}^n}\) itself and, under certain conditions on both \(A\) and \(\beta\), it can also be endowed with the structure of a complex mixed Hodge module. The main result of the paper states that, when \(A\) is homogeneous and normal (i.e., that \((1,\ldots,1)\) lies in the row span of \(A\) and \(\mathbb{N}A=\mathbb{Z}^d\cap\mathbb{R}_{>0}A\), respectively) and \(\beta\) lies within certain admissible region of \(\mathbb{C}^d\), the Hodge and order filtrations on \(\mathcal{M}_A^\beta\) coincide up to a shift, namely \(F_\bullet^H\mathcal{M}_A^\beta=F_{\bullet+d}^{\text{ord}}\mathcal{M}_A^\beta\).

In order to achieve this result, the authors generalise a path that was explored in some of their previous papers. Let \(h_A:(\mathbb{C}^*)^d\rightarrow\mathbb{C}^n\) be the torus embedding sending \(\underline{t}\) to \((\underline{t}^{a_1},\ldots,\underline{t}^{a_n})\), the \(a_i\) being the columns of \(A\). For a generic parameter vector \(\beta\) (namely a non-strongly resonant parameter of \(A\)), the direct image by \(h_A\) of certain simple twist of \(\mathcal{O}_{(\mathbb{C}^*)^d}\) is isomorphic to the Fourier-Laplace transform \(\check{\mathcal{M}}_A^\beta\) of \(\mathcal{M}_A^\beta\), so such object underlies a mixed Hodge module too. After a deep usage of Hodge module theory and some combinatorial properties of GKZ systems, the authors show in Section 4 that \(F_\bullet^H\check{\mathcal{M}}_A^\beta=F_{\bullet+d-n}^{\text{ord}}\check{\mathcal{M}}_A^\beta\).

The next step is to relate the Fourier-Laplace transformation with another integral transformation that generalises the Radon transformation, known in this case to preserve the mixed Hodge module structure, and keep trace of the behaviour of the Hodge filtration under all the functors involved in such integral transformation. This is done in the fifth section of the paper, in which every subsection is a step of the proof of the main theorem, that uses a skillful thread of results and calculations belonging to \(\mathcal{D}\)-module theory, homological algebra or semigroup rings. They also prove in a direct way a result due to Batyrev on the Hodge filtration on the relative cohomology of smooth affine hypersurfaces in algebraic tori.

In the last section of the paper, the authors use the main theorem to show that the reduced quantum \(\mathcal{D}\)-module of a nef complete intersection in a smooth projective toric variety underlies a variation of pure polarised non-commutative Hodge structures in the form of an \(\mathcal{R}\)-module.

Let \(A\) be a \(d\times n\) integer matrix with \(\mathbb{Z}A=\mathbb{Z}^d\) and \(\beta\in\mathbb{C}^d\) be a parameter vector. Take \(\mathbb{C}=\operatorname{Spec}\mathbb{C}[\lambda_1,\ldots,\lambda_n]\) and let \(\mathbb{L}\) be the kernel of \(A\), thought of as a map from \(\mathbb{Z}^n\) to \(\mathbb{Z}^d\). With this notation, define the GKZ system \(\mathcal{M}_A^\beta\) as the cyclic \(\mathcal{D}_{\mathbb{C}^n}\)-module given by \(\mathcal{D}_{\mathbb{C}^n}/\mathcal{I}_A\), where \(\mathcal{I}_A\) is the left ideal generated by the Euler operators \(\sum_j a_{ij}\lambda_j\partial_{\lambda_j}-\beta_i\) for \(i=1,\ldots,d\), and the toric operators \(\prod_{i:l_i>0}\partial_{\lambda_i}^{l_i}-\prod_{i:l_i<0}\partial_{\lambda_i}^{-l_i}\) for every \(l\in\mathbb{L}\).

Such a \(\mathcal{D}\)-module inherits the order filtration from \(\mathcal{D}_{\mathbb{C}^n}\) itself and, under certain conditions on both \(A\) and \(\beta\), it can also be endowed with the structure of a complex mixed Hodge module. The main result of the paper states that, when \(A\) is homogeneous and normal (i.e., that \((1,\ldots,1)\) lies in the row span of \(A\) and \(\mathbb{N}A=\mathbb{Z}^d\cap\mathbb{R}_{>0}A\), respectively) and \(\beta\) lies within certain admissible region of \(\mathbb{C}^d\), the Hodge and order filtrations on \(\mathcal{M}_A^\beta\) coincide up to a shift, namely \(F_\bullet^H\mathcal{M}_A^\beta=F_{\bullet+d}^{\text{ord}}\mathcal{M}_A^\beta\).

In order to achieve this result, the authors generalise a path that was explored in some of their previous papers. Let \(h_A:(\mathbb{C}^*)^d\rightarrow\mathbb{C}^n\) be the torus embedding sending \(\underline{t}\) to \((\underline{t}^{a_1},\ldots,\underline{t}^{a_n})\), the \(a_i\) being the columns of \(A\). For a generic parameter vector \(\beta\) (namely a non-strongly resonant parameter of \(A\)), the direct image by \(h_A\) of certain simple twist of \(\mathcal{O}_{(\mathbb{C}^*)^d}\) is isomorphic to the Fourier-Laplace transform \(\check{\mathcal{M}}_A^\beta\) of \(\mathcal{M}_A^\beta\), so such object underlies a mixed Hodge module too. After a deep usage of Hodge module theory and some combinatorial properties of GKZ systems, the authors show in Section 4 that \(F_\bullet^H\check{\mathcal{M}}_A^\beta=F_{\bullet+d-n}^{\text{ord}}\check{\mathcal{M}}_A^\beta\).

The next step is to relate the Fourier-Laplace transformation with another integral transformation that generalises the Radon transformation, known in this case to preserve the mixed Hodge module structure, and keep trace of the behaviour of the Hodge filtration under all the functors involved in such integral transformation. This is done in the fifth section of the paper, in which every subsection is a step of the proof of the main theorem, that uses a skillful thread of results and calculations belonging to \(\mathcal{D}\)-module theory, homological algebra or semigroup rings. They also prove in a direct way a result due to Batyrev on the Hodge filtration on the relative cohomology of smooth affine hypersurfaces in algebraic tori.

In the last section of the paper, the authors use the main theorem to show that the reduced quantum \(\mathcal{D}\)-module of a nef complete intersection in a smooth projective toric variety underlies a variation of pure polarised non-commutative Hodge structures in the form of an \(\mathcal{R}\)-module.

Reviewer: Alberto Castaño Domínguez (Sevilla)

##### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |