Let G be a torus and M a compact Hamiltonian G-manifold with finite fixed point set M ^G . If T is a circle subgroup of G with M ^G =M ^T , the T-moment map is a Morse function. We will show that the associated Morse stratification of M by unstable manifolds gives one a canonical basis of K _G (M). A key ingredient in our proof is the notion of local index I _p (a) for a∈K _G (M) and p∈M ^G . We will show that corresponding to this stratification there is a basis τ _p , p∈M ^G , for K _G (M) as a module over K _G (pt) characterized by the property: I _q τ _p =δ ^q _p . For M a GKM manifold we give an explicit construction of these τ _p 's in terms of the associated GKM graph.

In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry, namely the complete classification of strongly convex Randers metrics of constant flag curvature.

The Margulis invariant α is a function on H ¹(Γ,ℝ^2,1), where Γ is a group of Lorentzian transformations acting on ℝ^2,1, that contains no elliptic elements. The spectrum of Γ is the image of all γ∈Γ∖ (Id) under the map α. If the underlying linear group of Γ is fixed, Drumm and Goldman proved that the spectrum defines the translational part completely. In this note, we strengthen this result by showing that isospectrality holds for any free product of cyclic groups of given rank, up to conjugation in the group of affine transformations of ℝ^2,1, as long as it is non-radiant, and that its linear part is discrete and non-elementary. In particular, isospectrality holds when the linear part is a Schottky group.

In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollár's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita--Kawamata's semi-positivity theorem. The final section is an appendix, which is a result of Morihiko Saito.