Indices of 1-forms and quadratic forms

The index of an isolated singular point of a vector field or of a $1$-form on a smooth manifold can be expressed as the dimension of a certain algebra in the complex analytic setting and as the signature of a quadratic form on its real analogues (part) in the real analytic setting. This quadratic form has a natural definition in terms of residues. We shall discuss indices of singular points of $1$-forms on isolated complete intersection singularities and quadratic forms related to them.