A vector bundle is a family of vector spaces of fixed rank parametrized by a complex manifold. When trying to collect all vector bundles parametrized by the same compact complex manifold, one faces the problem that this collection is too big to have the structure of a compact complex manifold itself. The solution is to fix some invariants on the vector bundles (their Chern classes), and impose a numerical condition called stability.
In the surface case, there is a quadratic form (the discriminant) that is positive on the Chern classes of a stable vector bundle. In this case, it is possible to deform the vector bundles (to complexes of vector bundles) and the numerical stability condition so that the discriminant remains positive.
A similar quadratic form was conjectured to be positive on the Chern classes of stable complexes on threefolds. In this talk, after gently introducing some of the ideas above, I will present a counterexample for this conjecture on -dimensional blowups.