The eta-invariant of variation structures I

The main motivation of the paper is the signature of the Milnor fiber of the germ ƒ1 + ƒq2, where (ƒ1, ƒ2) is an isolated complete intersection singularity. This is computed in terms of a new invariant of the pair (ƒ1, ƒ2), which corresponds to the algebraic version of the eta-invariant, introduced by Atiyah, Patodi and Singer, for a hermitian flat bundle over the circle.