Integral representations for solutions of exponential Gauß-Manin systems

Abstract

Let f, g : U -> A(1) be two regular functions from the smooth affine complex variety U to the affine line. The associated exponential Gauss-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system O(U)e(g) with respect to f. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.