In this note we consider radially symmetric plurisubharmonic functions and the complex Monge-Ampère operator. We prove among other things a complete characterization of unitary invariant measures for which there exists a solution of the complex Monge-Ampère equation in the set of radially symmetric plurisubharmonic functions. Furthermore, we prove in contrast to the general case that the complex Monge-Ampère operator is continuous on the set of radially symmetric plurisubharmonic functions. Finally...

Let $X\subset {ℝ}^n$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$.

Let $X\subset {{ℂ}^n}^n$ be a closed real-analytic subset and put$$𝒜:=\{z\in X\mid \exists A\subset X,\text{germ}\text{of}\text{a}\text{complex-analytic}\text{set,}z\in A,{dim}_{z}A\>0\}$$This article deals with the question of the structure of $𝒜$. In the main result a natural proof is given for the fact, that $𝒜$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited.

This paper presents several theorems on the rectilinearization of functions definable by a convergent Weierstrass system, as well as their applications to decomposition into special cubes and quantifier elimination.

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let $\Phi :X\to Y$ be a morphism of real analytic spaces, and let $\Psi :𝒢\to ℱ$ be a homomorphism of coherent modules over the induced ring homomorphism ${\Phi }^{*}:{𝒪}_Y\to {𝒪}_X$. We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ${ℛ}_a=\mathrm{Ker}{\widehat{\Psi }}_a$, $a\in X$, are upper semi-continuous in the analytic Zariski topology of $X$. We prove semicontinuity in many cases (e.g. in the algebraic category)....

This is a sequel to “Relations among analytic functions I”, Ann. Inst. Fourier, 37, fasc. 1, [pp. 187-239]. We reduce to semicontinuity of local invariants the problem of finding ${𝒞}^{\infty }$ solutions to systems of equations involving division and composition by analytic functions. We prove semicontinuity in several general cases : in the algebraic category, for “regular” mappings, and for module homomorphisms over a finite mapping.

Given a real analytic manifold Y, denote by $Y_{sa}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $ℱ\mapsto {ℱ}^S$ on the category of sheaves on $Y_{sa}$ and study its properties. Roughly speaking, ${ℱ}^S$ is a sheaf on $X_{sa}\times S$. As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.

We give a characterization of the relative tangent cone of an analytic curve and an analytic set with an improper isolated intersection. Moreover, we present an effective computation of the intersection multiplicity of a curve and a set with s-metrization.

This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7, 8, 9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be...

We show an explicit relation between the Chow form and the Grothendieck residue; and we clarify the role that the residue can play in the intersection theory besides its role in the division problem.