Sections with isolated singularities

Let $ E\longrightarrow M $ be a holomorphic rank $ n $ vector bundle over a compact Kähler manifold of dimension $ n $, having a positive (or ample) line bundle $ L\longrightarrow M $ and consider a global section $ s $, with isolated singularities, of the twisted bundle $ E\otimes L^{\otimes r} $, where $ r $ is an integer. We prove that if $ r $ is large enough, then $ s $ is uniquely determined, up to a global endomorphism of the bundle $ E $, by its subscheme of singular points (which we call the singular subscheme of $ s $). If in particular $ E $ is simple, then $ s $ is uniquely determined, up to a scalar factor, by its singular subscheme. We recall that the last statement holds in case $ s $ is a holomorphic foliation by curves, with isolated singularities, on a projective manifold $ M $ with stable tangent bundle, so it holds in particular if $ M $ is a compact irreducible Hermitian symmetric space or a Calabi-Yau manifold. If $ L \longrightarrow \mathbb{P}^n $ is the hyperplane bundle, we show that it holds for every $ r\geq 1 $.

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Published 1 January 2003