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.

中文翻译：

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除数的部分密度函数的渐近性。

我们研究了与沿固定除数Y的特定阶消失的正厄米线束的部分相关的部分密度函数的渐近行为。假设所讨论的数据在\（S ^ 1 \）作用下（在Y周围）是不变的，我们证明了该密度函数具有分布渐近展开，实际上在传递到适当的实爆后它是平滑的。此外，我们恢复了密度函数呈指数减小的“禁区” R的存在，并证明其在边界\（\ partial R \）上具有“误差函数”行为。。作为说明性应用，我们使用它来研究可以与Kähler流形中的除数相关的某些自然函数。