This paper is concerned with singular projective rationally connected threefolds $X$ which carry non-zero pluri-forms, that is the reflexive hull of $({\rm\Omega}_{X}^{1})^{\otimes m}$ has a non-zero global section for some positive integer $m$. If $X$ has $\mathbb{Q}$-factorial terminal singularities, then we show that there is a fibration $p$ from $X$ to $\mathbb{P}^{1}$. Moreover, we give a formula for the numbers of $m$-pluri-forms as a function of the ramification of the fibration $p$.

We study subalgebras $A$ of affine or local algebras $B$ such that $A{\hookrightarrow}B$ is a pure extension from algebraic and geometric viewpoints.

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen–Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.

A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristics, including the wild case, was recently formulated by the author in the case where the given finite group acts linearly on an affine space. In cases of very special groups and representations, the conjecture has been verified and related stringy invariants have been explicitly computed. In this paper, we try to generalize the conjecture and computations to more complicated situations such as nonlinear actions on possibly singular spaces and nonpermutation representations of nonabelian groups.

We prove that the extension problem from one-dimensional subvarieties with values in Bergman space $H^{1}(D)$ on convex finite type domains can be solved by means of appropriate measures. We obtain also almost optimal results concerning the extension problem for other Bergman spaces and one-dimensional varieties.

We study the complete Kähler–Einstein metric of certain Hartogs domains ${\rm\Omega}_{s}$ over bounded homogeneous domains in $\mathbb{C}^{n}$. The generating function of the Kähler–Einstein metric satisfies a complex Monge–Ampère equation with Dirichlet boundary condition. We reduce the Monge–Ampère equation to an ordinary differential equation and solve it explicitly when we take the parameter $s$ for some critical value. This generalizes previous results when the base is either the Euclidean unit ball or a bounded symmetric domain.

First published online by Duke University Press 15 January 2010, subsequently published online by Cambridge University Press 11 January 2016, doi:10.1017/S0027763000009818