International Journal of Mathematics

We build compact moduli spaces of Grassmannian-framed bundles over a Riemann surface, essentially replacing a group by a bi-equivariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin–Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles (in the sense that all of the moduli can be obtained as quotients), and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

We study Lauricella's hypergeometric function F_C by using twisted (co)homology groups. We construct twisted cycles with respect to an Euler-type integral representation of F_C. These cycles correspond to 2^m linearly independent solutions to the system of differential equations annihilating F_C. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's F_C.

In this paper, by using the atomic decomposition theory of Hardy space H¹(ℝⁿ) and weak Hardy space WH¹(ℝⁿ), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

We prove that the maximum of two smooth strictly plurisubharmonic functions on an almost complex manifold can be uniformly approximated by smooth strictly plurisubharmonic functions.

Using the Białynicki-Birula method, we determine the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (5, 1, 4), (7, 2, 6), respectively, (0, 5, 19). We compute the Hodge numbers of these moduli spaces.

In a previous paper the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this paper, an analytic proof of this fact is given.