In $D$-Modules, Perverse Sheaves and Representation Theory from R. Hotta, K. Takeuchi and T. Tanisaki, I found the following statement (in section 8.2, the lines before Definition 8.2.2):

Setting: Let $X$ be an analytic space, $U\subset X$ Zariski-open, $j\colon U\hookrightarrow X$ the open embedding, $D^b_c(U)$ the (complexes of) sheaves with constructible cohomology, and $F\in D^b_c(U)$. Then there is stated basically the following:

If $\sqcup _{\alpha\in A}X_\alpha$ is a stratification of $X$ such that $F$ and $\mathrm{D}_U(F)$ (the dual of $F$) have locally constant cohomologies on each stratum $X_\alpha$, and one assumes that the given stratification satisfies the Whitney-conditions, then $Rj_\ast (F)\vert_{X_\alpha}$ and $j_!(F)\vert_{X_\alpha}$ have locally constant cohomologies again, so in particular $Rj_\ast(F),j_!(F)\in D^b_c(X)$.

Unfortunately, I don't see why this would be obviuos, so my question is:

Could anyone tell me a reference for this? As this is actually the first time I heard of the term "Whitney stratification", I would appreciate very much a reference that goes into some details.

Otherwise, if the statement should be trivial (sorry for my question in this case), any explanation on that would help me out a lot as well.

Thank you very much in advance!


1 Answer 1


For a discussion of Whitney stratifications, see for example Stratified Morse theory Goresky and Macpherson, Notes on topological stability by Mather, or Stratifications de Whitney et théorème de Bertini-Sard by Verdier. The point is that the conditions imply that everything is topologically locally trivial along strata in a suitably strong sense; constructibility of direct images follows pretty easily from this. To be a bit more explicit, assume for example, that $F$ is locally constant and that $X-U=X_\alpha$ (one can reduce to this case). Then locally the inclusion $j:U\to X$ is homeomorphic to $X_\alpha\times (L\cap U)\subset X_\alpha \times L$ for a suitable transversal slice $L$. Then $R^ij_*F|_U$ is $F$ when $i=0$ and $0$ otherwise, and $R^i j_*F|_{X_\alpha}$ is the locally constant sheaf associated to $H^i(L\cap U, F)$, so $Rj_*F$ is constructible.

  • $\begingroup$ Thanks a lot!! (and special thanks for pointing out the idea of what to use the Whitney conditions for!) $\endgroup$
    – user103697
    Feb 25, 2017 at 22:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.