Noether-Lefschetz problems for degeneracy loci - Introduction

In this monograph we study the cohomology of degeneracy loci of the following type. Let psi: F --> E be a general homomorphism between holomorphic vector bundles E and F on a projective complex manifold X. We assume that E circle times F-v is ample (see Definition 2.1.3) and generated by its global sections. Fix an integer r. Then we are interested in Z := D-r(psi) := {x is an element of X\ rk psi(x) less than or equal to r}. We set n := dim(X), e := rk(E) and f := rk(F). Replacing psi with its transpose, if necessary, we assume without loss of generality that f less than or equal to e.