Abstract
We 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 partial 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.
Highlights
This is independent of choice of orthonormal basis, and our interest lies in its asymptotic behaviour as k tends to infinity
In this paper we study the analogous partial density function associated to sections of high powers of a positive hermitian line bundle that vanish to a particular order along a fixed divisor
Speaking it states that there is a natural way in which the partial density function has a globally defined asymptotic expansion in powers of k1/2 whose terms depend on the curvature of the hermitian metric, all of which are computable
Summary
The density function of a hermitian line bundle is the restriction to the diagonal of the Bergman kernel, which is the reproducing kernel for the L2-projection to the space of holomorphic sections. We shall discuss the simple extension of this concept in which we impose a certain vanishing of the sections along a fixed submanifold
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