Asymptotic integrals and Hodge structures

The survey is devoted to the asymptotics of integrals of the method of descent and to the Hodge structures of critical points of phases of integrals. The problem of how asymptotics change under deformation of phases is discussed.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 22, pp. 130–166, 1983.