On stability of projections of Lagrangian varieties

We show that Lagrangian and Legendre varieties associated with matrix singularities and singularities of composite functions are stable in a sense which is a natural modification of Givental’s notion of stability of Lagrangian projections. The study of singular Lagrangian and Legendre varieties was initiated about twenty five years ago by Arnold when he was investigating singularities in the variational problem of obstacle bypassing [1]. The first examples of such varieties, open swallowtails, were related to the discriminants of the non-crystallographic Coxeter groups [4, 8]. Incorporating these examples into a general context, Givental [4] introduced the notion of stability of Lagrangian and Legendre varieties with respect to perturbations of symplectic structure and Lagrangian or, respectively, contact structure and Legendre projection only, keeping the diffeomorphic type of the variety fixed. Later, in [7], it was shown that this stability notion has an explicit geometrical meaning in terms of generating families, versal deformations of function singularities and inducing mappings. The interest in theory of singular Lagrangian and Legendre varieties has been growing recently due to its possible applications to Frobenius structures, D-modules and other areas. In this paper we extend the results of [7] to a natural modification of Givental’s stability notion and show that the stability condition holds for a wide class of Lagrangian and Legendre varieties associated with matrix singularities (see [2, 3, 6]) and singularities of composite functions [5].