Suppose $X$ is a smooth algebraic variety over a field of characteristic $0$. What are the most general conditions under which Hochschild homology and cohomology of $X$ agree? The existence of a symplectic form will do the job, but is there something more general? I would be interested in particular into a condition along the lines of Lurie's recent approach to isomorphism of homology and cohomology via the theory of ambidexterous functors (iso between a left and a right adjoint to a given functor, under some specific conditions).
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$\begingroup$ If I understand correctly, the ambidexterity condition that Hopkins and Lurie study (which is a property rather than a structure on a variety) would only apply when your variety is not just CalabiYau but zerodimensional CalabiYau (giving the isomorphism in Sasha's answer but with no shift). $\endgroup$– David BenZviSep 30, 2012 at 18:45
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If $X$ is CalabiYau (that is $K_X = 0$) then Hochschild homology and cohomology agree up to a shift of grading by dimension of $X$.