Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

Homogeneous Finsler spaces and the flag-wise positively curved condition

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space {S^{3}\times\mathbb{R}} which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on {S^{2}\times S^{3}} and {S^{6}\times S^{7}} which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on {S^{3}\times S^{3}} , shedding some light on the long standing general Hopf conjecture.

THE HOPF CONJECTURE FOR MANIFOLDS WITH ABELIAN GROUP ACTIONS

Let M be a closed even n-manifold of positive sectional curvature on which a torus Tk acts isometrically. We show that if [Formula: see text] (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each Tk-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group [Formula: see text]-action on a closed manifold of positive sectional curvature.