an:07189072
Zbl 1445.14033
Fang, Jiangxue
Composition series for GKZ-systems
EN
Trans. Am. Math. Soc. 373, No. 5, 3445-3481 (2020).
00448415
2020
j
14F10 32S60
GKZ system; \(\mathcal D\)-module
\textit{V. V. Batyrev} [Duke Math. J. 69, No. 2, 349--409 (1993; Zbl 0812.14035)], \textit{J. Stienstra} [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Singapore: World Scientific. 412--452 (1998; Zbl 0963.14017)], and \textit{A. Adolphson} [Duke Math. J. 73, No. 2, 269--290 (1994; Zbl 0804.33013)] considered an increasing filtration \(W_{\bullet}(A,\beta)\) on the GKZ system \(M_{A}(\beta)\). When \(\beta\) is special, this filtration is related to the mixed Hodge structure of the cohomology of hypersurfaces in a toric variety.
In the article under review, the author studies the question that whether the associated graded pieces of this filtration are semisimple \(\mathcal D\)-modules. The author constructs canonical epimorphisms from these associated graded pieces to some \(\mathcal D\)-modules coming from ``smaller'' GKZ systems, and gives a criterion when these epimorphisms can be made isomorphisms. In particular, he shows that if \(A\) is ``simplicial relative to \(\beta\)'' and \(\beta\) is ``weakly \(A\)-nonresonant'' (two conditions that are combinatorial in nature), then these associated graded pieces are indeed semisimple \(\mathcal D\)-modules.
Dingxin Zhang (Beijing)
Zbl 0812.14035; Zbl 0963.14017; Zbl 0804.33013