Existence Of Vector Bundle On Complex Surface With Special Properties

Given the set Z of isolated point and a cohomology class c∈H2(S, Z)on the complex surface S, what we ask is whether there exist a rank two holomorphic vector bundle Eâ†'S on S such that its first Chern class is the given cohomology class and it has a global sections∈H0(S, O(E)) whose divisor is Z. We denote I be the regular ideal sheaf associate to Z, and L be a line bundle with Chern class is the given cohomology class. This paper draws that if we can find an elemente∈Ext1(S;I,L)such that for each p∈Z,ep is a unit of Ext(I, L)pthis problem can be solved. Then this paper got that if H2(S,L)=0,we have a solution for this problem. When L is the sheaf O of germs of holomorphic functions and every stalk o I is the maximal ideal of Oz,we got that this problem can be solved if and only if there exist a bi-vectors0≠τp∈∧2Tp(S),(p∈Z) such that∑〈Ψ,Ï„p〉=0for each Ψ∈H0(S,Ω2), and we proved that this condition is equivalent to that satisfy the Caley-bacharch property.