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).



When do Hochschild homology and cohomology agree? (Ambidexterity?)
If $X$ is Calabi--Yau (that is $K\_X = 0$) then Hochschild homology and cohomology agree up to a shift of grading by dimension of $X$.


