On the Mathematical Work of Professor Heisuke Hironaka

In this succinct and incomplete presentation of Hironaka’s published work up to now, it seems convenient to use a covering according to a few main topics: families of spaces and equisingularity, birational and bimeromorphic geometry, finite determinacy and algebraicity problems, flatness and flattening, real analytic and subanalytic geometry. This order follows roughly the order of publication of the first paper in each topic. One common thread is the frequent use of blowing-ups to simplify the algebraic problems or the geometry. For example, in the theory of subanalytic spaces of R, Hironaka inaugurated and systematically used this technique, in contrast with the “traditional” method of studying subsets of R by considering their generic linear projections to Rn−1. No attempt has been made to point at generalizations, simplifications, applications, or any sort of mathematical descent of Hironaka’s work, since the result of such an attempt must be either totally inadequate or of book length.