International Journal of Mathematics

Let P be a self-adjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form , where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions.

(1) The restriction π|_H is discretely decomposable in the sense of Kobayashi.

(2) The momentum map is proper.

In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

We study the twisted Alexander polynomial from the viewpoint of the SL(2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL(2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g - 2.

In the present paper we classify curves and surfaces in Euclidean 3-space which make constant angle with a certain Killing vector field. Moreover, we characterize the catenoid and Dini's surface in terms of constant angle surfaces.

Let V be a strongly regular vertex operator algebra. For a state h ∈ V₁ satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr_M q^L(0)-c/24ζ^h(0) (M a V-module) is a vector-valued weak Jacobi form of weight 0 and a certain index 〈h, h〉/2. We discuss refinements and applications of this result when V is holomorphic, in particular we prove that if g = e^h(0) is a finite-order automorphism then Tr_V q^L(0)-c/24g is a modular function of weight 0 on a congruence subgroup of SL₂(ℤ).

In this paper, some kinds of random shadowing properties are introduced. It is shown that, for some natural analogs of the deterministic case, a random hyperbolic set satisfies the shadowing properties.

We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of Cⁿ. This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls.

In this paper, we study a special class of Finsler metrics which are defined by a Riemannian metric and a 1-form on a manifold. We find equations that characterize Ricci-flat Douglas metrics among this class.

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (and Riemannian real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor.

In this note, we consider locally invertible analytic mappings of a two-dimensional space over a non-archimedean field. Such a map is called semi-hyperbolic if its Jacobian has eigenvalues λ₁ and λ₂ so that λ₁ = 1 and |λ₂| ≠ 1. We prove that two analytic semi-hyperbolic maps are analytically equivalent if and only if they are formally equivalent, applying a generalized version of an estimation scheme from our earlier work [A. Jenkins and S. Spallone, A p-adic approach to local dynamics: Analytic flows and analytic maps tangent to the identity, Ann. Fac. Sci. Toulouse Math. (6) 18(3) (2009) 611–634].

The stable singularities of differential map germs constitute the main source of studying the geometric and topological behavior of these maps. In particular, one interesting problem is to find formulae which allow us to count the isolated stable singularities which appear in the discriminant of a stable deformation of a finitely determined map germ. Mond and Pellikaan showed how the Fitting ideals are related to such singularities and obtain a formula to count the number of ordinary triple points in map germs from ℂ² to ℂ³, in terms of the Fitting ideals associated with the discriminant.

In this article we consider map germs from (ℂ^n+m, 0) to (ℂ^m, 0), and obtain results to count the number of isolated singularities by means of the dimension of some associated algebras to the Fitting ideals. First in Corollary 4.5 we provide a way to compute the total sum of these singularities. In Proposition 4.9, for m = 3 we show how to compute the number of ordinary triple points. In Corollary 4.10 and with f of co-rank one, we show a way to compute the number of points formed by the intersection between a germ of a cuspidal edge and a germ of a plane.

Furthermore, we show in some examples how to calculate the number of isolated singularities using these results.

In this paper the relationship between the Ricci curvature and the fundamental groups of Finsler manifolds are studied. We give an estimate of the first Betti number of a compact Finsler manifold. Two finiteness theorems for fundamental groups of compact Finsler manifolds are proved. Moreover, the growth of fundamental groups of Finsler manifolds with almost-nonnegative Ricci curvature are considered.

Let Ω be a bounded hyperconvex domain in ℂⁿ, 0 ∈ Ω, and S_ε a family of N poles in Ω, all tending to 0 as ε tends to 0. To each S_ε we associate its vanishing ideal and pluricomplex Green function .

Suppose that, as ε tends to 0, converges to (local uniform convergence), and that (G_ε)_ε converges to G, locally uniformly away from 0; then . If the Hilbert–Samuel multiplicity of is strictly larger than its length (codimension, equal to N here), then (G_ε)_ε cannot converge to . Conversely, if is a complete intersection ideal, then (G_ε)_ε converges to . We work out the case of three poles.

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊_n supported by 𝕊_n are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊_n supported by A to be finite dimensional.