Real deformations and invariants of map-germs

Abstract

Abstract. A stable deformation f t of a real map-germ f : R n , 0 → R p , 0 is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification f t C are real. A related notion is that of a good real perturbation f t of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of f t coincides with that of f t C . The class of map germs having an M-deformation is, in some sense, much larger than the one having a good real perturbation. We show that all singular map-germs of minimal corank (i.e. of corank max(n − p + 1, 1)) and A e -codimension 1 have an M-deformation. More generally, there is the question whether all A-simple singular map-germs of minimal corank have an M-deformation. The answer is "yes" for the following three dimension ranges (n, p): n ≥ p, p ≥ 2n and p = n + 1, n = 4. We describe some new techniques for obtaining these results, which lead to simpler proofs and also to new results in the dimension range n + 2 ≤ p ≤ 2n − 1.

M-deformations of -simple germs from to

“…For general (n, p) it is known that every A e -codimension 1 orbit of singular map-germs C n → C p of minimal corank has a real representative which in turn has a good real perturbation [5,14,21]. And it is also known that every real A e -codimension 1 singular map-germ of minimal corank has an M-deformation [30] -notice that the second statement holds for a larger class of map-germs: for example, the complex A-orbit of f = (x, y 3 + x 2 y) has representatives f ± = (x, y 3 ± x 2 y) (distinct over the reals), both having an M-deformation, but only f + (the lip) has a good real perturbation (not the beak-to-beak f − ).…”