Mathematical Research Letters 10, 651–658 (2003) ON SECTIONS WITH ISOLATED SINGULARITIES OF TWISTED BUNDLES AND APPLICATIONS TO FOLIATIONS BY CURVES Antonio Campillo and Jorge Olivares Dedicated to the memory of A.N. Tyurin Abstract. Let E -→ M be a holomorphic rank n vector bundle over a compact K¨ahler manifold of dimension n, having a positive (or ample) line bundle L -→ M and consider a global section s, with isolated singularities, of the twisted bundle E ⊗ L ⊗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 -→ P n is the hyperplane bundle, we show that it holds for every r ≥ 1. 1. Introduction In the previous paper [5], the authors have shown that an algebraic foliation of degree at least 2 in the projective plane is determined by the subscheme of its singular points (or its singular subscheme, after Definition 2.1 below). This result was known to hold for foliations by curves in projective spaces having a reduced singular subscheme (see [8]). In this paper we study our previuos result in a general context, providing the following extension: A foliation by curves (with isolated singularities) of a sufficiently high degree on a compact projective manifold whose tangent bundle is simple, is determined by its singular subscheme (Corollary 3.2). Since stable bundles are simple (Proposition 2.5 below), this holds in partic- ular for the manifolds studied in [19] and [22]. We shall show (also in Corollary 3.2) that it holds as well for the classes of the so-called compact irreducible Her- mitian symmetric spaces (see [10] or [1] for the definition and classification of Received November 21, 2003. 2000 Mathematics Subject Clasification. Primary 32S65; Secondary 32L10. Partially supported by DGCYT PB97-0471. Partially supported by CONACYT I29879-E and AECI - CCI (Mexico). Thanks to A.N. Tyurin, X. G´omez-Mont and A.N. Todorov for useful conversations. 651