International Journal of Mathematics

We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over to a compactified universal Picard.

In this paper, we consider logarithmic transformations for torus bundles over a nonsingular algebraic curve and give a criterion to be Kähler or projective for torus quasi bundles. In paticular we have a criterion to be Kähler or projective for elliptic surfaces with c₂=0.

We show that on an arbitrary nilpotent orbit in where is a direct sum of classical simple Lie algebras, there is a G-invariant hyperKähler structure obtainable as a hyperKäher quotient of the flat hyperKähler manifold ℝ^4N≅ℍ^N. Coïncidences between various low-dimensional simple Lie groups lead to some nilpotent orbits being described as hyperKähler quotients (in some cases in fact finite quotients) of other nilpotent orbits. For example, from the construction we are able to read off pairs of orbits in different classical Lie algebras such that there is a finite -equivariant surjection between the orbit closures. We include a table listing examples of hyperKähler quotients between small nilpotent orbits.

The above-mentioned results have consequences in quaternionic Kähler geometry: it is known that nilpotent orbits in complex semisimple Lie algebras give rise to quaternionic Kähler manifolds. Our approach gives a more direct proof of this in the classical case as these manifolds turn out to be quaternionic Kähler quotients of quaternionic projective spaces. We find that many of these manifolds can also be constructed as quaternionic Kähler quotients of complex Grassmannians .

Carrying further the work of T.A. Crawford, we show that each component of the space of harmonic maps from the 2-sphere to complex projective 2-space of degree d and energy 4πE is a smooth closed submanifold of the space of all C^j maps (j≥2). We achieve this by showing that the Gauss transform which relates them to spaces of holomorphic maps of given degree and ramification index is smooth and has injective differential.

For any complex manifold of Kähler type, the L²-norm of the scalar curvature of an extremal Kähler metric is a continuous function of the Kähler class. In particular, if a convergent sequence of Kähler classes are represented by extremal Kähler metrics, the corresponding sequence of L²-norms of the scalar curvatures is convergent. Similarly, the sequence of holomorphic vector fields associated with a sequence of extremal Kähler metrics with converging Kähler classes is convergent.

For each invertible m×m matrix Q a compact matrix quantum group A_u(Q) is constructed. These quantum groups are shown to be universal in the sense that any compact matrix quantum group is a quantum subgroup of some of them. Their orthogonal version A_o(Q) is also constructed. Finally, we discuss related constructions in the literature.

Let X be a -analytic space which is not necessarily of finite dimension and let us assume that X is holomorphically spreadable. In this paper, we prove that, in this situation, X is q-complete (resp. cohomologically q-complete) if and only if X is q-convex (resp. cohomologically q-convex) for any integer q≥1. Also, analogous problems within the framework of holomorphically convex spaces will be discussed.