International Journal of Mathematics

We study properties of a Green function G_A with singularities along a complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u≤ log |ψ|+C, where ψ=(ψ₁,…,ψ_m), ψ₁,…,ψ_m are local generators for the ideal sheaf ℐ_A of A, and C is a constant depending on the function u and the generators. A motivation for this study is to estimate global bounded functions from the sheaf ℐ_A and thus proving a "Schwarz lemma" for ℐ_A.

We study the structure of complex points on real surfaces, embedded into complex elliptic surfaces. We show, for example, that any compact connected surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc bundles over compact connected orientable surfaces that have a Stein structure as Stein domains inside elliptic surfaces.

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.

Let X be a reduced pure n-dimensional complex analytic set in ℂ^N with an isolated singularity at 0. We obtain some L²-existence and regularity results for the -operator on a deleted neighborhood of the singular point 0 in X.

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.

The purpose of this article is twofold. The first is to prove the unicity theorem with truncated multiplicities of meromorphic functions sharing five small functions. This gives a remarkable improvement of the results of Yuhua–Jianyong, Yao and Yi. The second is to generalize the unicity theorem of Fujimoto to meromorphic functions sharing four small functions with truncated multiplicities.