International Journal of Mathematics

We prove a double coset formula for induced representations of compact Lie groups. We apply it to the representation rings of unitary and symplectic groups to obtain Hopf algebras. We also construct a Heisenberg algebra representation based on the restiction and induction of representations of unitary groups.

We demonstrate the uniqueness of an antiholomorphic involution on a certain class of Stein manifolds. Examples are given to show why this uniqueness no longer holds without some topological assumptions.

A wedge of k circles is homotopy equivalent to the plane with k points removed. By looking at ∖{w₁, …, w_k} instead of ⋁^k S¹, we could view the symmetric product of a wedge of k circles as , the complement space [3] of a hyperplane arrangement, . Applying a theorem of Hattori [2], we see that if k > n, the symmetric product of ⋁^k S¹ is, up to homotopy, the union of n-dimensional subtorii in a k-torus.

In this paper an analytic proof of a generalization of a theorem of Bismut [1, Theorem 5.1] is given, which says that, when v is a transversal holomorphic vector field on a compact complex manifold X with a zero point set Y, the embedding j:Y→ X induces a natural isomorphism between the holomorphic equivariant cohomology of X via v with coefficients in ξ and the Dolbeault cohomology of Y with coefficients in ξ|_Y, where ξ→ X is a holomorphic vector bundle over X.

We study the canonical stability of a smooth projective 3-fold V of general type. We prove that |5K_V| gives a birational map onto its image provided p_g ≥ 4. For those V with p_g = 3, |6K_V| gives a birational map. Known examples show that both the results are optimal.

The lifting invariants of a closed subscheme X ⊂ Pⁿ are the numbers , where H is a general hyperplane and where f is the restriction map. The lifting invariants measure the obstruction to lift hypersurfaces (of H) containing X ∩ H to hypersurfaces containing X. We first prove a general result (which holds for every X ⊂ Pⁿ) on the behaviour of the r_i's; then we turn to the special case of space curves and, under some special assumptions, we prove vanishing results for the r_i's and for the cohomology.

Let V be a local complete intersection in a complex manifold W. For a function g on W, we set f = g|_V and f′ = g|_V′, where V′ denotes the non-singular part of V. For each compact connected component S of the union of the singular set of V and the critical set of f′, we define the virtual multiplicity of f at S as the residue of the localization by df′ of the Chern class of the virtual cotangent bundle of V. The multiplicity m(f, S) of f at S is then defined by , where μ(V, S) is the (generalized) Milnor number of [2]. If S = {p} is an isolated point and if g is holomorphic, we give an explicit expression of as a Grothendieck residue on V. In the global situation, where we have a holomorphic map of V onto a Riemann surface, we prove a singular version of a formule of B. Iversen [13].

In this paper we make the first steps to a classification of (pseudo-) Riemannian manifolds which are not locally homogeneous but their Levi–Civita connections are homogeneous. The full classification is given for dimension n = 2; in higher dimensions we prove some substantial partial results. In more generality, we are also interested in the difference between the dimension of the algebra of affine Killing vector fields and that of the algebra of metric Killing vector fields (without any homogeneity properties).