The reviewed paper concerns the following situation. Let \(f:({\mathbb K}^n,0)\to({\mathbb K}^p,0)\) be a smooth map-germ (\({\mathbb K}={\mathbb C}\) or \({\mathbb R}\), for \({\mathbb K}={\mathbb C}\) smooth means complex-analytic, for \({\mathbb K}={\mathbb R}\) smooth means either \(C^\infty\) or real-analytic). Such germs form the space \(A(n,p)\) [see J. W. Bruce and T. J. Gaffney, J. Lond. Math. Soc., II. Ser. 26, 465–474 (1982; Zbl 0575.58008); C. T. C. Wall, Bull. Lond. Math. Soc. 13, 481–539 (1981; Zbl 0451.58009)]. The group \({\mathcal R}\) (respectively, \({\mathcal L}\)) is the group of diffeomorphisms \(({\mathbb K}^n,0)\rightarrow({\mathbb K}^n,0)\) (respectively, \(({\mathbb K}^p,0)\rightarrow({\mathbb K}^p,0)\)). The group \({\mathcal A}={\mathcal R}\times{\mathcal L}\) acts on \(A(n,p)\) by \((\varphi,\psi)\cdot f=\psi\circ f\circ\varphi^{-1}\). Two germs \(f,g\in A(n,k)\) are said to be \(\mathcal A\) equivalent if they lie in the same \(\mathcal A\) orbit. The contact group \(\mathcal K\) is the group of germs of diffeomorphisms \(H:({\mathbb K}^{n+p},0)\rightarrow({\mathbb K}^{n+p},0)\) such that there exists a diffeomorphism \(h:({\mathbb K}^n,0)\rightarrow({\mathbb K}^n,0)\) which satisfies (i) \(H\circ i=i\circ h\), where \(i:({\mathbb K}^n,0)\rightarrow({\mathbb K}^n\times{\mathbb K}^p,0)\) is the inclusion \(i(x)=(x,0)\); and (ii) \(\pi\circ H=h\circ\pi\), where \(\pi:({\mathbb K}^n\times{\mathbb K}^p,0)\rightarrow({\mathbb K}^n,0)\) is the natural projection. The action \(H\cdot f\) on \(f\in A(n,p)\) is defined by \((h(x),H\cdot f(x))=H(x,f(x))\). Now \(f,g\in A(n,k)\) are \(\mathcal K\) equivalent if they lie in the same \(\mathcal K\) orbit.
A map-germ \(f\in A(n,k)\) is said to be \({\mathcal A}\) stable [see J. Damon and A. Galligo, Invent. Math. 32, 103–132 (1976; Zbl 0333.57017)] if for a representative of \(f\) defined in a neighbourhood of 0, \(f_1:U\rightarrow{\mathbb K}^p\), and a mapping \(f_2:U\rightarrow{\mathbb K}^p\) sufficiently near \(f_1\) in the Whitney topology (i.e., the topology of uniform convergence of all the derivatives on compact sets) there are germs of diffeomorphisms \(h:{\mathbb K}^n\rightarrow{\mathbb K}^n\) and \(k:{\mathbb K}^p\rightarrow{\mathbb K}^p\) but with \(h(0),k(0)\) possibly different from zero such that \(k^{-1}\circ f_2\circ h=f\) as a germ at 0. Let \(J^k(n,p)\) be the space of \(k\)-jets of mappings \(A(n,p)\) and for a fixed group \({\mathcal G}\) let \({\mathcal G}^k\) be the corresponding space of \(k\)-jets of diffeomorphisms. A singular map-germ \(f\in A(n,p)\) is said to be \({\mathcal G}\) simple if there exist \(K>0\) and \(L>0\) such that, for every \(k\geq K\), there exists a neigbourhood (in \(J^k(n,p)\)) of the orbit \(f{\mathcal G}^k\) that intersects not more than \(L\) other orbits [V. I. Arnold, et al. Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts. Monographs in Mathematics, Vol. 82. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001)] or S. Janeczko, Selected topics in catastrophe theory, in Polish].
The authors of the reviewed paper consider the following classification problems. (1) The classification of germs of smooth maps \(f:({\mathbb K}^n,0)\rightarrow({\mathbb K}^p,\Omega_p,0)\) up to \({\mathcal A}_{\Omega_p}\) equivalence (i.e., for the subgroup of \({\mathcal A}\) in which the left coordinate changes preserve a given volume form \(\Omega_p\) in the target \({\mathbb K}^p\)), and also of multi-germs of such maps up to \({\mathcal A}_{\Omega_p}\) equivalence. (2) The classification of variety-germs \(V=f^{-1(0)}\subset({\mathbb K}^n,\Omega_n,0)\) up to \({\mathcal K}_{\Omega_n}\) equivalence of \(f:({\mathbb K}^n,\Omega_n,0)\rightarrow({\mathbb K}^p,0)\) (i.e., for the subgroup of \({\mathcal K}\) in which the right coordinate changes preserve a given volume form \(\Omega_n\) in the source \({\mathbb K}^n\)). The authors study the \({\mathcal G}_{\Omega_q}\) moduli space of \(f\) that parametrizes the \({\mathcal G}_{\Omega_q}\) orbits (\(q=n\) or \(p\)) inside the \({\mathcal G}\) orbit of \(f\) (a discussion: moduli vs. parameter spaces is given for example in [J. Harris and I. Morrison, Moduli of curves. Graduate Texts in Mathematics. 187. New York, NY: Springer (1998; Zbl 0913.14005)]; the term “modality” is the counterpart of the term “number of parameters”). The authors find, for example, that this moduli space vanishes for \({\mathcal G}_{\Omega_q}={\mathcal A}_{\Omega_p}\) and \({\mathcal A}\) stable maps \(f\) and for \({\mathcal G}_{\Omega_q}={\mathcal K}_{\Omega_n}\) and \({\mathcal K}\) simple maps \(f\). On the other hand, they show that there are \({\mathcal A}\) stable maps \(f\) with infinite-dimensional \({\mathcal A}_{\Omega_n}\) moduli space.