We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand et al. [Generalized Euler integrals and -hypergeometric functions, Adv. Math. 84 (1990), 255-271] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves and twisted Morse theory are used.
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