Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle E over X. When pg(X) = 0, we construct deformations of holomorphic structures on E such that the new bundle is trivial over any ADE curve C inside C and therefore descends to the singular surface obtained by contracting C.

The problem of$subextension$of plurisubharmonic functions is considered. Recently it was shown by Cegrell–Zeriahi, that subextension is always possible for negative plurisubharmonic functions in the energy class ℱ. In this paper we construct, for every hyperconvex domain Ω, a negative plurisubharmonic function in the class ℰ which cannot be subextended. Given any pseudoconvex domain we construct a pluriharmonic function that cannot be subextended.

The main purpose of this paper is to study the asymptotic property of one class of number-theoretic functions and give a sharp hybrid mean-value formula by using the generalized Bernoulli numbers, Gauss sums and the mean-value theorems of Dirichlet$L$-functions.

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1. Initial Data Set formulation The initial data set is given by (gij, pij) on a three dimensional manifold where gij is asymptotically euclidean (so that asymptotic quantities can be defined) and pij falls off quadratically. We also study the case when g^ is asymptotically hyperbolic and the three manifold is asymptotically null. Several important questions remain unsolved for the initial data set and its Cauchy development.