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...

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.

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.