In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry.
For example, in a map from a connected complex surface to the complex projective line, a generic fiber is a smooth Riemann surface of some fixed genus g and, generically, there will be isolated points in the target whose preimages are nodal curves.
The loop in the smooth fibers gives an element of the first homology group of a surface, and the monodromy of the critical value is defined to be the monodromy of the first homology of the fibers as the loop is traversed, i.e. an invertible map of the first homology of a (real) surface of genus g. A classical result is the Picard–Lefschetz formula,[1] detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a shear mapping.
The classical, geometric theory of Solomon Lefschetz was recast in purely algebraic terms, in SGA7.
This formulation has been of continuing influence, in particular in D-module theory.