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Hello.

I am sorry if this is a totally trivial question. I am interested in computing cohomologies of polyvector fields of an arbitrary complete (algebraic) toric variety (assume smooth if otherwise it's becoming too complicated). I would be very grateful if you could help me with this. How can this be done? Do you have in mind any relevant references? Thank you!

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  • $\begingroup$ What is a polyvector field? $\endgroup$ Jun 30, 2012 at 14:09
  • $\begingroup$ By cohomologies of polyvector fields I mean taking cohomologies with coefficients in exterior powers of the tangent bundle. Thank you for asking. $\endgroup$
    – Maria Pap
    Jul 1, 2012 at 6:59
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    $\begingroup$ That sounds like a great question, and I am sure there must be an answer. Unfortunately I am not an expert. For every invertible sheaf $L$ on a smooth, projective toric variety $X$, the cohomology $H^q(X,\Omega^p_X\otimes L)$ forms a natural subspace of the appropriate weight space of $H^q(U,\Omega^p_U)$, where $U$ is the universal torsor; a particular open subset of affine space. The cohomology $H^q(U, \Omega^p_U)$ can be "computed" by local cohomology. In principle this might work. Also there is an analogue of the Euler sequence for $X$; that might work also. $\endgroup$ Jul 2, 2012 at 14:08
  • $\begingroup$ Thanks so much!I think I have some reading to do regarding your first suggestion. For the second one, I have the impression that the analogue of the Euler sequence works only for simplicial toric varieties (however I am not sure if the restriction of being simplicial is so important in my case). Thank you again :-) $\endgroup$
    – Maria Pap
    Jul 3, 2012 at 11:10
  • $\begingroup$ @Maria -- Also my first suggestion only works in the "simplicial" case, i.e., every open cone in the fan is the convex hull of a (rational or real) basis for the vector space. This is necessary for the universal torsor to be an open subset of an affine space, i.e., for the Cox ring to be a polynomial ring. $\endgroup$ Jul 3, 2012 at 14:48

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