On versality for unfoldings of smooth section germs
In recent years, the ''versality theorem'' for categories of unfoldings of smooth map germs relative to some equivalence relations has been proved (Martinet, Mather, Zakalyukin). Now, we say that the ''versality theorem'' holds if the algebraic notion of ''infinitesimal versality'' is the sufficient condition of the notion of ''versality''. But, as the category of unfoldings of smooth vector fields germs relative to coordinate transformations, there are examples for which the ''versality theorem'' cannot hold [see \textit{G. R. Belitskij,} Usp. Mat. Nauk 33, No.1(199), 95- 155 (1978; Zbl 0385.58007); translation from Russ. Math. Surv. 33, No.1, 107-177 (1978)]. In this paper, the author singles out a class of categories of unfoldings of smooth section germs of smooth vector bundles relative to various equivalence relations for which the ''versality theorem'' holds. As corollaries of the main theorem, the author shows the ''versality theorem'' for categories of unfoldings of smooth map germs (Martinet, Mather, Zakalyukin), and the category of G-unfoldings of G-invariant function germs relative to G-right equivalence [\textit{V. Poenaru}, Singularités \(C^{\infty}\) en présence de symétrie (Lect. Notes Math. 510) (1976; Zbl 0325.57008)]. Namely, the main theorem of this paper unifies these theorems. Moreover, it gives categories other than those stated above for which the ''versality theorem'' holds. For example, it is obtained that for the category of unfoldings of solutions of Poisson equations \(\Delta f=g\) the ''versality theorem'' holds.
scientific article; zbMATH DE number 3842805