ar X iv : m at h / 05 04 38 0 v 1 [ m at h . A G ] 1 9 A pr 2 00 5 Hypersurface Singularities and Milnor Equisingularity ∗ Lê

ar X iv : m at h / 04 11 43 2 v 4 [ m at h . A G ] 7 F eb 2 00 5 Hypersurface Singularities and the Swing ∗ Lê

Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of f . This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

ar X iv : m at h / 04 11 43 2 v 3 [ m at h . A G ] 2 3 N ov 2 00 4 Hypersurface Singularities and the Swing ∗ Lê

Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of f . This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

ar X iv : m at h / 05 04 37 3 v 1 [ m at h . Q A ] 1 9 A pr 2 00 5 Lax Operator for the Quantised Orthosymplectic

ar X iv : m at h / 04 09 02 9 v 1 [ m at h . A G ] 2 S ep 2 00 4 ACM BUNDLES ON GENERAL HYPERSURFACES IN P 5 OF LOW DEGREE

In this paper we show that on a general hypersurface of degree r = 3, 4, 5, 6 in P 5 a rank 2 vector bundle E splits if and only if h 1 E(n) = h 2 E(n) = 0 for all n ∈ Z. Similar results for r = 1, 2 were obtained in [15], [16] and [1].

ar X iv : m at h / 05 04 16 3 v 1 [ m at h . D G ] 8 A pr 2 00 5 AFFINE CURVATURE HOMOGENEOUS 3 - DIMENSIONAL LORENTZ MANIFOLDS

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous , but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing...