A note on Chow stability of the Projectivisation of Gieseker Stable Bundles

We investigate Chow stability of projective bundles P(E) where E is a strictly Gieseker stable bundle over a base manifold that has constant scalar curvature. We show that, for suitable polarisations L, the pair (P(E),L) is Chow stable and give examples for which it is not asymptotically Chow stable.Comment: 23 pages. Theorem 2 changed to fix an error in conventions, and the Example in Section 5 rewritten to accommodate. Other minor changes, mostly expositiona

Slope Stability and Exceptional Divisors of High Genus

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is slope stable but not K-stable.Comment: Published versio

Asymptotics of Partial Density Functions for Divisors.

This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s12220-016-9741-8We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor Y. Assuming the data in question is invariant under an S 1 -action (locally around Y) we prove that this density function has a distributional asymptotic expansion that is in fact smooth upon passing to a suitable real blow-up. Moreover we recover the existence of the "forbidden region" R on which the density function is exponentially small, and prove that it has an "error-function" behaviour across the boundary ∂ R . As an illustrative application, we use this to study a certain natural function that can be associated to a divisor in a Kähler manifold.Engineering and Physical Sciences Research Council (Career Acceleration Fellowship, Grant ID: EP/J002062/1), Leverhulme Trus