http://mathoverflow.net/questions/9492/how-does-one-find-vanishing-algebraic-cycles
Ayoub’s work.
arXiv:1002.1686 Notes on Beilinson’s “How to glue perverse sheaves” from arXiv Front: math.CT by Ryan Reich The titular, foundational work of Beilinson not only gives a technique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These constructions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also defines a new, “maximal extension” functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems.
math/9908107 Notes on Perverse Sheaves and Vanishing Cycles from arXiv Front: math.AG by David B. Massey 1 person liked this A working mathematician’s summary of many results on the derived category, perverse sheaves, and vanishing cycles. This is the February 2010 version.
nLab page on Vanishing cycles