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

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