Exceptional vector bundles associated to degenerations of surfaces -- Paul Hacking, March 6, 2009

If a smooth complex surface Y degenerates to a singular surface X with a ordinary double point x^2+y^2+z^2=0, then the specialisation map H_2(Y,Z) --> H_2(X,Z) has nontrivial kernel, generated by the so called vanishing cycle: a 2-sphere in Y which collapses to a point in X. However, in the theory of moduli of surfaces, more complicated degenerations naturally occur which have no vanishing cycles. In some of these cases we construct a rigid holomorphic vector bundle F on the smooth fibre Y which is analogous to a vanishing cycle. The case of the projective plane will be described explicitly.