On Vanishing Theorems for Local Systems Associated to Laurent Polynomials

Abstract

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-Kapranov-Zelevinsky [10] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves and twisted Morse theory will be used.

Local Systems on Complements of Arrangements of Smooth, Complex Algebraic Hypersurfaces

“…If A is an affine complex arrangement, work of Kohno [Koh86], Esnault, Schechtman, Varchenko [ESV92], and Schechtman, Terao, Varchenko [STV95] gives sufficient conditions for a local system L on M (A) to insure the vanishing of the cohomology groups H i (M (A), L) for all i < rank(A). Similar conditions for the vanishing of cohomology of with coefficients in rank 1 local systems were given by Levin and Varchenko [LV12] for elliptic arrangements, and by Esterov and Takeuchi [ET17] for certain toric hypersurface arrangements. In turn, we provide in Corollary 2.9 a unified set of generic vanishing conditions for cohomology of local systems on complements of arrangements of smooth, complex algebraic hypersurfaces.…”