A note on the Pullback Conjecture

The problem of whether the pullback of a singular analytic germ via a proper holomorphic self-map germ of $(\mathbb C^n,0)$ stays singular dates back to 2007 and has not been solved yet, apart from a number of special cases. In this paper we start with a brief overview of the main advances made so far. Then, we observe that analogous results can be proved globally, for algebraic sets and a proper polynomial map. Secondly, we study the case of proper map germs $\mathbb C^n \to \mathbb C^m$, i.e. with $m\geq n$.

We also present some additional results, the first of which is an application of a slightly generalised version of the Giraldo–Roeder reduction. Another one involves Lipschitz geometry and Lipschitz normally embedded sets.