Edward Bierstone

Semianalytic and subanalytic sets

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I. The Tarski-Seidenberg theorem and Thorns lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Semianalytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. Subanalytic sets 16 4. Transforming an analytic function to normal crossings by blowings-up 21 5. Uniformization and rcctilincarization 30 6. I.ojasicwiczs inequality; metric properties of subanalytic sets 33 7. Smooth points of a subanalytic sct 37 Bibliography 42

A simple constructive proof of Canonical Resolution of Singularities

In these notes, we describe some of the main features of an explicit proof of canonical desingularization (of algebraic varieties or analytic spaces X) in characteristic zero. Full details will appear in [7]. The proof is a variation on our proof of local desingularization (“uniformization”) [4], [5], and justifies the philosophy that “a sufficiently good local choice [of centre of blowing-up] should globalize automatically” [5, p. 901]. The final version is surprisingly elementary; these notes, for example, include an essentially self-contained presentation of the hypersurface case. The general case involves a “reduction to the hypersurface case” result from [5].

Geometric and differential properties of subanalytic sets

We announce solutions of two fundamental problems in differential analysis and real analytic geometry, on composite differentiable functions and on semicoherence of subanalytic sets. Our main theorem asserts that the problems are equivalent and gives several natural necessary and sufficient conditions in terms of semicontinuity of discrete local invariants and metric properties of a closed subanalytic set.

Desingularization of toric and binomial varieties

We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness criterion of Hironaka. The results extend to more general varieties defined locally by binomial equations.

Geometric Auslander criterion for flatness

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let

Local analytic invariants and splitting theorems in differential analysis

\varphi: X \to Y

Arc-quasianalytic functions

be a morphism of complex-analytic spaces, where

Geometric Auslander criterion for openness of an algebraic morphism

Y

Solutions of quasianalytic equations

is smooth. We prove that nonflatness of

Desingularization by blowings-up avoiding simple normal crossings

\varphi