Let g be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra U(g) has intermediate growth and thus infinite Gelfand–Kirillov (GK-) dimension. We prove that the GK-dimension of U(g) is just infinite in the sense that any proper quotient of U(g) has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra S(g) by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space X such that every run of the K_X-MMP immediately terminates with a Mori fibre space, yet X does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.

We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset U of a metric space X, the predecessor U_∗ of U is defined by doubling the radii of all open balls contained inside U, and taking their union. The predecessor of U is an open set containing U. The directed doubling distance between U and another subset V is the number of times that the predecessor operation needs to be applied to U to obtain a set that contains V. Finally, the doubling distance between open sets U and V is the maximum of the directed distance between U and V and the directed distance between V and U.

In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.

In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen–Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.