## Relative canonical sheaf

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Hi.

I want to know if for $f:X\to S$ a proper flat holomorphic map with n-dimensionnal fibers over reduced complex space S, the relative canonical sheaf $w_{X/S}:=H^{-n}(f^{!}O_{S})$ is a dualizing sheaf which imply that the two functor, on Coh(S), $G\to H^{-n}(f^{!}G)$ and $G \to f^{*}G\otimes w_{X/S}$ agree...

Thanks.

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1
Thank you for the answer. But, it seems to me that a relative dualizing sheaf is necessarly flat over S and then compatible with any base change (Kleiman, "Relative duality.. Compositio math, 41, prop 9 p.9). We know that for cohen macaulay morphism (conrad: Grothendieck duality) or morphism with Dubois singularities ( Kollar-Kovaks, journal.Am.Math.soc. 23, p.791) it is the case. In general, the recent paper of Zsolt :Arxiv 28 may 2010 " base change for relative canonical sheaf in families of normal varieties) show us that it is no true. – kaddar Jun 11 2010 at 10:43

## 2 Answers

If the relative canonical sheaf $w_{X/S}:=H^{-n}(f^{!}O_{S})$ is a dualizing sheaf then necessarily the two functors $G-->H^{-n}(f^{!}G)$ and $G-->f^{*}G\otimes w_{X/S}$ agree as well.

Remark that the first functor is left exact and the second is right exact which imply immediately that $w_{X/S}$ is flat over $S$ and then commute with any base change!

But it is not true in general. For this, we can see the simple example of Zsolt in Arxiv AG-2008: Base change for relative canonical sheaf. In this paper, he consider a flat family of normal varieties over smooth curve.

I have another very easy example which show us that $w_{X/S}$ is compatible with base change but the two functor dont agree. This example is giving by a surjective finite morphism between Normal Gorentein complex spaces. Of course, this morphism is neither flat, neither of finite tor dimension but it defines an analytic family of zero-cycles (and then we have a nice holomorphic trace map).

Important remark: 1) In this two example, Kunz relative sheaf (relative regular meromorphic forms) and relative canonical sheaf agree.

2) Kunz relative sheaf is not a dualizing sheaf in general.

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If $f\colon X \to S$ a proper flat of schemes map with $n$-dimensionnal fibers over a noetherian scheme $S$, the relative canonical sheaf $\omega_{X/S}:=H^n(f^!{\mathcal{O}_S})$ is a dualizing sheaf. I guess that this should imply what you want by a GAGA-type argument.

Indeed, being $f$ flat we have that $f^!G = f^*G \otimes f^!{\mathcal{O}_S}$ [Lipman, SLN 1960, Theorem 4.9.4] for $G \in D^b_c(X)$ (or, more generally, for $G \in D^+_{qc}(X)$). Now, $f^!{\mathcal{O}_S}$ is concentrated between degrees 0 and $-n$ by looking at the fibers of $f$ and its description via residual complexes. Then, taking $-n$-th cohomology on both sides and one obtains the desired result.

Unfortunately, I am not familiar enough with the analytic version of the story as in Ramis-Ruget-Verdier "Dualité relative en géométrie analytique complexe", but I guess the algebraic version may give you a clue for how to transpose the result to your setting. I would bet you do not need $S$ reduced as long as $f$ is flat.

Addendum

There was a confusion on my part. I was speaking about the dualizing complex while the original question was about the dualizing sheaf. If we agree that $\omega_X = H^{-n}(f^!O_B)$ for a equidimensional family (with $n$ the dimension of the fibers), then the base change isomorphism in the derived category for a square with base $u \colon P \to B$, ($B$ for "base scheme") a base change of the map $f \colon X \to B$ completing the square with $v \colon F \to X$ and $g \colon F \to P$, respectively.

Then, by the formula in the derived category,

$Lv^* f^! G \cong g^! Lu^* G$

we get an isomorphism of sheaves

$H^{-n}(Lv^* f^!O_B) \cong H^{-n}(g^! Lu^*O_B) = H^{-n}(g'^!O_P)$

so we see that the dualizing sheaf over $F$ is the abutment of a spectral sequence involving higher $Tor$ sheaves related to the embedding of the fiber into the space $X$. But, of course one needs some information on this embedding to get the collapse of the spectral sequence.

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