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Where can I find references on semismall maps, in the sense of Goresky and MacPherson? I don't want to restrict to the case where the base is $\mathbb C$ (an arbitrary alg. closed field would be fine), or maps $f:X\to Y$ from a smooth variety $X.$ In particular, I'd like to find the proof (if the statement is correct, which I'm not sure) that $Rf_*$ takes an irreducible (middle) perverse sheaf $F$ supported on $X$ to a perverse sheaf; I can only do this when $X$ is smooth and $F$ is a lisse sheaf, or when all the fibers of $f$ have dimension at most one.

Recall: A proper surjective morphism $f:X\to Y$ is called $semismall$ if $\dim X\times_YX=\dim X.$

Thank you.

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Have you seen "Weil conjectures, Perverse sheaves, and l'adic Fourier transform" chapter 3? There (III.7.5) the assertion is proved for smooth perverse sheaves, though it is not fully what you want. –  Akhil Mathew Aug 14 '11 at 22:19
    
Thanks for giving the reference. I don't have access to the book right now. Do you mind copying the statement of III.7.5 for me? If it's for the case when $X$ is smooth and $F$ is lisse, I think I know how to prove it. By the way, I'm not assuming the existence of a stratification (as in KW); this should be included in the proof, if the statement is correct. –  shenghao Aug 15 '11 at 10:01
    
The statement is, Suppose $X$ smooth and equidim. of dim. $n$ over the base field $k$. Suppose $f: X \to Y$ proper and let $A = \mathcal{G}[n]$ a smooth perverse sheaf on $X$ (i.e. $\mathcal{G}$ is a smooth etale sheaf on $X$). Then if $f$ is semi-small, $Rf_* A \in \mathrm{Perv}(Y)$. If $f$ is small, then $Rf_* A = j_{!*} (Rf_* A|U)$ (where $U$ is one part of the stratification). –  Akhil Mathew Aug 15 '11 at 15:23
    
P.S. I recently saw your paper on generalizations of the Grothendieck trace formula to algebraic stacks. I found it very interesting! –  Akhil Mathew Aug 15 '11 at 15:23
    
This (that $Rf_*IC(L)$ is a perverse sheaf, for a local system $L$ defined on some open set) seems to hold when fibers of $f$ have dimension at most 1. In general it might be false, as ulrish said in the following. –  shenghao Aug 28 '11 at 20:30

1 Answer 1

The statement that $Rf_*$ takes a middle perverse sheaf to a perverse sheaf is not true for arbitrary perverse sheaves:

For example, let $Y$ be a smooth surface and $X$ the blow up of $Y$ at a point. Let $E$ be the exceptional divisor and $F$ the constant sheaf on $E$ placed in degree $-1$ (to make it perverse). The natural morphism $f:X \to Y$ is small but $Rf_* F$ is not perverse since it has cohomology sheaves supported on a point in degrees $-1$ and $1$.

EDIT: The other counterexample, suggested in a previous edit, does not appear to work.

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Thanks for pointing it out; I realized that I was sloppy about it too. I meant to say "perverse sheaf with support $X$". I'm not sure if it's true or not. –  shenghao Aug 15 '11 at 9:58
    
Maybe you want "irreducible perverse sheaf with support $X$"? Otherwise one could take the direct sum of any perverse sheaf with $IC_X$... –  ulrich Aug 15 '11 at 10:17
    
@ulrich: Could you show me how to compute the cohomology groups in the counter-example you removed, please? Precisely, $R\Gamma(E,IC_X(L)|_E)$ and the stalk of $IC_Y(L)$ at the origin. Thanks. –  shenghao Aug 18 '11 at 14:14
    
@shenghao: One can compute everything explicitly using Deligne's construction of $IC$ sheaves. The only non-formal ingredient is the fact that the cohomology of a non-trivial rank $1$ local system on $\mathbb{A}^1 - \{0\}$ is trivial in all degrees. Both $IC$ sheaves turn out to be just $i_*$ of the local system and $G$ is in fact $0$. –  ulrich Aug 19 '11 at 9:29
    
@shenghao: If one analyzes the proof for lisse sheaves, it can be seen that what you want holds if $dim(X) \leq 2$. To construct counterexamples (which I am told do exist) one needs to have a local system $L$ on an open subset of $X$ so that $IC(L)$ has non-zero cohomology sheaves (with respect to the usual $t$-structure) in more degrees than just $- \ dim(X)$. –  ulrich Aug 19 '11 at 9:36

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