Flatness in analytic mappings. I: On an announcement of Hironaka (Q804760)

In the research announcement in Proc. Japan Acad., Ser. A 62, 73-75 (1986; Zbl 0589.32015), \textit{H. Hironaka} asserts the existence of a ``semicoherent'' stratification for a subanalytic subset. The sketch of the proof contains three steps. In the paper under review the authors prove that the accouncement of Hironaka is not correct. In fact they prove the first step, namely: Theorem. Let \(\Phi: Y\to M\) be a proper morphism of real analytic spaces with M smooth and \(X=\Phi (Y)\) (which is closed and subanalytic). Let \({\mathcal F}\) be a coherent sheaf of \({\mathcal O}_ Y\)-modules. Then there exists a locally finite stratification of X, say \(X=\cup_{i}X_ i\), such that for each \(T=X_ i:\) (1) T is a locally closed real analytic submanifold of M which is connected and subanalytic in M. (2) Let \({\mathcal H}_ T\) denote the ideal sheaf in \({\mathcal O}_ Y\) (within a small neighborhood of \(\Phi^{-1}(T))\) generated by the ideal sheaf of T in \({\mathcal O}_ M\) via \(\Phi\). Then the restriction to \(\Phi^{-1}(T)\) of \(\oplus^{\infty}_{k=0}{\mathcal H}^ k_ T\cdot {\mathcal F}/{\mathcal H}_ T^{k+1}\cdot {\mathcal F}\) is flat as a \({\mathcal O}_ T\)-module. However, by counterexamples, it is shown that both step 2 and 3 in Hironaka's announcement are not correct. The proof of the theorem above is based on a generic normal flatness result which is proved using semicontinuity properties of the ``diagram of initial exponents'' and also properties of the parametrized families formal of power series (earlier developed by the authors).