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Let $X$ be a non-smooth toric variety, $\Omega_X$ be the sheaf of differentials, $\hat{\Omega}_X$ the double dual of $\Omega_X$. My questions are:

  1. Is there any chance that $\Omega_X=\hat{\Omega}_X$?
  2. If 1 is not true, is there any chance that we can relate the cohomology of the two sheaves?
  3. If 1 and 2 does not have an answer in general, hwo about weighted projective spaces?
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    $\begingroup$ This doesn't quite answer your question, but it's related. See the main theorems in arXiv:1003.2913. Keeping in mind that normal toric varieties are always Kawamata log terminal. In particular, the pushdown of the sheaf of differentials from a resolution of a toric variety should satisfy this property. S\'andor will likely give more specifics. $\endgroup$
    – Karl Schwede
    Commented Nov 23, 2010 at 21:27
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    $\begingroup$ Just a quick comment, in some sources the sheaf $\hat{\Omega}_X$ is called the ``Zariski-de Rham differentials''. This might help you find more sources.. $\endgroup$
    – Karl Schwede
    Commented Nov 24, 2010 at 16:37
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    $\begingroup$ @Karl, I always wondered if these have an "official" name. We usually just call them reflexive differentials. $\endgroup$
    – Sándor Kovács
    Commented Nov 24, 2010 at 20:44

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1) No, this usually does not happen. Take the singularity $x^2+y^2+z^2=0$. Then $$xdx+ ydy+zdz=0$$ and hence one obtains that $$\eta:=\frac{xdz-zdx}y=\frac{zdy-ydz}x.$$ Now this implies that $\eta$ is a differential form that is defined on the complement of the singular point and hence it is in the reflexive hull, but it cannot be extended into the singular point. So $\Omega_X$ is not reflexive.

2/3) There is of course the natural map $\Omega_X\to \hat\Omega_X$, so there is a map on cohomology, but

4) Why would you want to know the cohomology of $\Omega_X$? The interesting sheaf here is the reflexive hull of $\Omega_X$, that is, in your notation, $\hat\Omega_X$. As Karl points out that has the nice property that it is isomorphic to $\pi_*\Omega_{\widetilde X}$ if $\pi:\widetilde X \to X$ is a resolution of singularities. It also is the sheaf whose cohomology appears in the Hodge structure of these singularities. (Along with the reflexive hulls of $\Omega_X^p$ which are also isomorphic to $\pi_*\Omega_{\widetilde X}^p$ by the paper Karl quoted above).

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  • $\begingroup$ Well, it is kind of a long story why I am interested in $\Omega_X$ rather than its reflexsive hull. Roughly speaking because the deformation theory is controlled by $\Omega_X$. I am trying to understand the deformation of a pair $(X, D)$. A question I asked a few days ago is here: mathoverflow.net/questions/45343/… The divisor $D$ is isomorphic to a (rather singular, but normal) weighted hypersurface in a weighted projective space. (But $X$ is not the weighted projective space!) So I am trying to relate the cohomology of D to the w.p.s. $\endgroup$
    – Zhiyu
    Commented Nov 24, 2010 at 15:17
  • $\begingroup$ It seems like there is an upper bound on how many words one can use in a comment. Let me explain a little more. Basically I need an isomorphism between $H^i(Ext(\Omega_D, \mathcal{O}_D)$ to $H^{i+2}(Hom(\Omega_D, \mathcal{O}_D))$. One way to do this is to try to relate these to the ambient weighted projective space. $\endgroup$
    – Zhiyu
    Commented Nov 24, 2010 at 15:37
  • $\begingroup$ I was going to say (after your first comment) that you need $Ext$ and not the cohomology of $\Omega$, which is a trivial comment, but explains why I asked what I asked in the answer... $\endgroup$
    – Sándor Kovács
    Commented Nov 24, 2010 at 16:06

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