Return to Question

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at vanishing cycles, nearby cycles, and specialization? I have a decent idea how some of it works for studying the cohomology of a one-parameter flat family of degenerating complex manifolds (see below), but the general sheaf picture still gives me a headache. Any recognition principles (e.g., "this looks like a place where I can use a vanishing cycle argument") would be most welcome.

Say I have a family of complex manifolds where the fibers are smooth over a punctured unit disc, and have some mild singularities over zero (a priori the singularities could be arbitrarily bad, but say we blow up until we have simple normal crossings). The cohomology of the fibers forms a vector bundle on the punctured disc, and it comes equipped with some extra structure, such as a pure Hodge filtration and a Gauss-Manin connection that identifies nearby fibers. When we attempt to extend the vector bundle over the whole disc, the extra structures degenerate - the Hodge structure becomes "mixed", and the connection acquires logarithmic singularities. These structures aren't immediately relevant to this question, but they seem to be interesting.

As far as I can tell, vanishing cycles and nearby cycles arise when we try to relate the cohomology of smooth fibers X_t with that of the special fiber X₀. Each smooth fiber X_t has an inclusion map to the total space X, and X is homotopy equivalent to X₀ by a fiberwise retraction. The composition yields a map from X_t to X₀, and the pushforward of a sheaf on X_t along this map yields the nearby cycles sheaf. When I start with the constant sheaf on X_t this yields a sheaf on X₀ that computes cohomology of X_t for some abstract nonsense reason. So far, I'm okay, but it seems that choosing t is not canonical enough, so one replaces X_t with the homotopy equivalent universal fiber X_oo over the universal cover of the punctured disc (the upper half plane), and defines nearby cycles by some crazy pullback-pushforward-pullback sequence. Specialization and vanishing cycles seem to be similar - I think there is a nice geometric picture somewhere, but the proliferation of upper and lower stars make me sad. Is there a good way to see through that thicket?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at vanishing cycles, nearby cycles, and specialization? I have a decent idea how some of it works for studying the cohomology of a one-parameter flat family of degenerating complex manifolds (see below), but the general sheaf picture still gives me a headache. Any recognition principles (e.g., "this looks like a place where I can use a vanishing cycle argument") would be most welcome.

Say I have a family of complex manifolds where the fibers are smooth over a punctured unit disc, and have some mild singularities over zero (a priori the singularities could be arbitrarily bad, but say we blow up until we have simple normal crossings). The cohomology of the fibers forms a vector bundle on the punctured disc, and it comes equipped with some extra structure, such as a pure Hodge filtration and a Gauss-Manin connection that identifies nearby fibers. When we attempt to extend the vector bundle over the whole disc, the extra structures degenerate - the Hodge structure becomes "mixed", and the connection acquires logarithmic singularities. These structures aren't immediately relevant to this question, but they seem to be interesting.

As far as I can tell, vanishing cycles and nearby cycles arise when we try to relate the cohomology of smooth fibers X_t with that of the special fiber X₀. Each smooth fiber X_t has an inclusion map to the total space X, and X is homotopy equivalent to X₀ by a fiberwise retraction. The composition yields a map from X_t to X₀, and the pushforward of a sheaf on X_t along this map yields the nearby cycles sheaf. When I start with the constant sheaf on X_t this yields a sheaf on X₀ that computes cohomology of X_t for some abstract nonsense reason. So far, I'm okay, but it seems that choosing t is not canonical enough, so one replaces X_t with the homotopy equivalent universal fiber X_oo over the universal cover of the punctured disc (the upper half plane), and defines nearby cycles by some crazy pullback-pushforward-pullback sequence. Specialization and vanishing cycles seem to be similar - I think there is a nice geometric picture somewhere, but the proliferation of upper and lower stars makes me sad. Is there a good way to see through that thicket?