The vanishing cohomology of non‐isolated hypersurface singularities

Abstract

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it to compute explicitly the lowest (possibly nontrivial) vanishing cohomology group of the Milnor fiber.

Milnor fibers of homogeneous polynomials of prime power degree

“…The point of this section to introduce terminology and notation required to state and understand the statement of the vanishing cycle result in Theorem 3.4.a of Maxim, Pȃunescu, and Tibȃr from [9]. However, while discussing monodromy, we also recall the statements of the Monodromy Theorem and A'Campo's Theorem on the Lefschetz number of the Milnor monodromy.…”

“…Now, we give the result from Theorem 3.4.a of [9], which is obtained by using the cosupport condition satisfied by the perverse sheaf of shifted vanishing cycles along f . Then there is an injection…”

“…We denote the invariant Z-submodule of this monodromy representation by inv F f,C • . By a result of Siersma ([10]) in the case s = 1 and, more generally, by a result of Maxim, Pȃunescu, and Tibȃr ( [9]; recalled in Theorem 2.6) using the cosupport condition for the perverse sheaf of shifted vanishing cycles, the cohomology at the Milnor fiber at the origin is not independent of the cohomology of the nearby Milnor fibers; in fact, H n−s (F f,0 ; Z) injects into 2010 Mathematics Subject Classification. 32B15, 32C18, 32B10, 32S25, 32S15, 32S55.…”

“…We wish to thank Laurentiu Maxim for many comments, and in particular for providing a reference to [9] for the vanishing cycle result in Theorem 2.6. We also thank Laurentiu for pointing out Steenbrink's proof in 6.4 of [11] of the result which is referred to in this paper as Theorem 3.3.…”

A note on the support and cosupport conditions for a perverse sheaf

“…It is our hope that these new support and cosupport conditions will become as equally wellknown as the two versions given in Definition 1.1, mainly for the new version of the cosupport condition. We, ourselves, and others have instead had to essentially prove that (C2) implies the new cosupport condition in the midst of other results; this is the case, for instance, in the proof of Theorem 3.1 in [5] where A • is the perverse sheaf of shifted vanishing cycles along a function f .…”