Mixed polynomials are polynomials in complex variables u and v as well as their complex conjugates ū and v. They are therefore identical to the set of real polynomial maps from to . We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials to mixed polynomials, introducing the concepts of partially non-degenerate and strongly partially non-degenerate mixed functions. We prove that partial non-degeneracy implies the existence of a weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity. We also compare (strong) partial non-degeneracy with other types of non-degeneracy of mixed functions, such as (strong) inner non-degeneracy, and find that, in contrast to the holomorphic setting, the different properties are not equivalent for mixed polynomials. We then define classes of mixed polynomials for which strong partial non-degeneracy is equivalent to the existence of an isolated singularity. Furthermore, we prove that mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition, resulting in an explicit Milnor (sphere) fibration.

In a curvature-adapted hypersurface M of a quaternionic-Kähler manifold M the maximal quaternionic subbundle of TM and its orthogonal complement in TM are invariant subspaces of the shape operator at each point. We classify curvature-adapted real hypersurfaces M of non-flat quaternionic space forms and that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where in the hyperbolic case we assume additionally that the hypersurace has constant principal curvatures. The position vector of a such submanifold in the ambient (pseudo) Euclidean space is decomposable into a sum of a constant vector and two nonconstant vector eigenfunctions of the Laplace operator of the submanifold belonging to different eigenspaces. In the quaternionic projective space they include geodesic hyperspheres of arbitrary radius r ∈ (0, π/2) except one, two series of tubes about canonically embedded quaternionic projective spaces of lower dimensions and two particular tubes about a canonically embedded . On the other hand, the list of 2-type curvature-adapted hypersurfaces with constant principal curvatures in is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic quaternionic hyperplane . Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere H₃ in is not of finite type but satisfies = const.

Given a finite group G, we define a new invariant of odd-dimensional oriented closed manifolds and call it the KDW invariant. This invariant is a Dijkgraaf-Witten invariant in terms of K-theory. In this paper, we compute the invariant of the Brieskorn homology spheres with G = We should remark that, in this computational result, the fundamental groups of the Brieskorn homology spheres and are not nilpotent.

This is a continuation of the authors' earlier work on deformations of cuspidal edges. We give a representation formula for swallowtails in the Euclidean 3-space. Using this, we investigate map-germs of generic swallowtails in 3-dimensional space from, and show some important properties of them. In particular, we give a representation formula giving all map germs of swallowtails in the Euclidean 3-space whose Gaussian curvatures are bounded from below by a positive constant or by a negative constant from above. Using this, we show that any swallowtails are deformed into a swallowtail of constant Gaussian curvature in the space form preserving the sign of their Gaussian curvatures.

Paolo Aluffi showed that the Chern-Schwartz-MacPherson class of the complement of a free arrangement agrees with the total Chern class of the sheaf of logarithmic derivations along the arrangement. We describe the defect of equality of the two classes for locally tame arrangements with isolated non-free singular loci.