Ramifications of Holomorphic Integrals

Abstract

We consider the integral

$${\rm I}({\rm x})=\int\limits_{{\rm S}_{\rm q}({\rm x})}{\cal U}({\rm x},\tau){\rm d}\tau\ ;$$

I(x) is holomorphic and ramified around V : φ(x, τ) = 0 ; we integrate on the relative cycle defined by the holomorphic simplex S_q (x) and its faces. We obtain the ramification of I(x) using discriminants and polar manifolds. In fact, I(x) is ramified around a hypersurface : Δ(x) = 0 ; denote V_x = {τ; φ(x, τ) = 0} ; if x is such that Δ(x) = 0, either V_x has a singular point, or an edge of the simplex is tangent to V_x . Assumptions are essentially Weierstrass hypothesises on functions induced by φ on grassmannian manifolds.