The coefficients of series expansions of integrals, the weight and Hodge filtrations and the spectrum of a critical point

Let us consider the integral of a holomorphic differential form over a homology class of a continuous family of integral homologies of fibres of the Milnor fibration of a critical point. The function given by the integral can be expanded in a series of powers of the parameter and powers of the logarithm of the parameter of the family (Chapter 10). Each coefficient of the series depends linearly both on the form and on the continuous family of integral homologies. If the form is fixed but the continuous family varies, then each coefficient of the series is a linear function of the continuous families. Linear combinations, over ℂ of continuous families of integral homology classes form the space of covariantly constant (with respect to the Gauss-Manin connection) sections of the homological Milnor fibration. Therefore (if the form is fixed) each coefficient of the series is a linear function on the space of covariantly constant sections of the homological Milnor fibration, that is it is a covariantly constant section of the cohomological Milnor fibration.