On vanishing theorems for local systems associated to Laurent polynomials. (English) Zbl 1408.14161
Let \(B=\{ b(1), b(2), \ldots ,b(N)\} \subset \mathbb{Z}^{n-1}\) be a finite subset of the lattice \(\mathbb{Z}^{n-1}\) such that the affine lattice generated by \(B\) in \(\mathbb{Z}^{n-1}\) coincides with \(\mathbb{Z}^{n-1}\). The authors consider Laurent polynomials \(P(x)=\sum_{j=1}^Nz_jx^{b(j)}\) and for \(c=(c_1,\ldots ,c_n)\in \mathbb{C}^n\), possibly multivalued functions of the form \(P(x)^{-c_n}x_1^{c_1-1}\cdots x_{n-1}^{c_{n-1}-1}\). These generate rank-one local systems on \((\mathbb{C}^*)^{n-1}\setminus P^{-1}(0)\). The authors extend results of I. M. Gelfand et al. [Adv. Math. 84, No. 2, 255–271 (1990; Zbl 0741.33011)] about concentration of cohomology in middle degrees. In the course of the proof, some properties of vanishing cycles of perverse sheaves and twisted Morse theory are used.
Reviewer: Vladimir P. Kostov (Nice)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 32S22 | Relations with arrangements of hyperplanes |
| 32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
| 33C70 | Other hypergeometric functions and integrals in several variables |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
Citations:
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