I'm really happy and proud to have Hochschild cohomology of Hilbert schemes of points on surfaces on the arXiv now, joint with my good friends Lie Fu and Andreas Krug.

There is a (to me at least) funny story behind the paper. In Huybrechts's famous book on Fourier–Mukai transforms, he briefly discusses Hochschild (co)homology through a larger and bigraded algebra which contains both Hochschild cohomology and Hochschild homology, and also the canonical algebra, following Orlov's presentation in his survey paper. It is defined for a smooth projective variety $X$ as \begin{equation*} \operatorname{HS}_*^\bullet(X) = \bigoplus_{i\in\mathbb{Z}}\bigoplus_{j\in\mathbb{Z}} \operatorname{HS}_j^i(X) = \bigoplus_{i\in\mathbb{Z}}\bigoplus_{j\in\mathbb{Z}} \operatorname{Ext}_{X\times X}^i(\Delta_*\mathcal{O}_X,\Delta_*\omega_X^{\otimes j}[j\dim X]), \end{equation*} where we have (for a good reason) inserted a certain shift when compared to the original definition. To avoid a conflict of terminology, and emphasise the role of the Serre functor in this definition, we have decided to call this algebra the Hochschild–Serre cohomology of $X$. Because this bigraded algebra is a derived invariant, those 3 invariants are also derived invariants.

But, nowhere had I ever seen other applications of this bigraded algebra. Was it really useful on its own, or only as a vehicle for those 3 invariants?

The Hochschild homology of Hilbert schemes of points on surfaces is easy to compute (as a graded vector space) because we have Göttsche's formula for the Hodge numbers of Hilbert schemes (as implemented in the Hodge diamond cutter, for instance). Thus, using the Hochschild–Kostant–Rosenberg decomposition, we can leverage this information to know the dimension of Hochschild homology.

The Hochschild cohomology, however is a different beast. Sure, we have the Hochschild–Kostant–Rosenberg decomposition, which involves exterior powers of the tangent bundle now. But getting a good enough grasp on the tangent bundle of the Hilbert scheme seems like a challenging problem.

Two ingredients

This is where Lie, Andreas, and I tried using the famous derived McKay correspondence due to Bridgeland–King–Reid–Haiman equivalence, which gives a derived equivalence \begin{equation*} \mathbf{D}^{\mathrm{b}}(\operatorname{Hilb}^nS) \cong \mathbf{D}^{\mathrm{b}}([\operatorname{Sym}^nS]) \end{equation*} where on the right-hand side we have the derived category of the symmetric quotient stack. And Hochschild cohomology is a derived invariant, so we can compute it using the right-hand side.

The second ingredient is the orbifold Hochschild–Kostant–Rosenberg decomposition due to Arinkin, Căldăraru, and Hablicsek. This can be applied to the right-hand side of the McKay correspondence.

And this is where the magic happens! When you work out the orbifold decomposition of the Hochschild cohomology, you suddenly see the Hochschild–Serre cohomology appear. The reason is that the computation involves fixed loci, and thus various diagonals, which when you spell out the details can be packaged together into one formula which reads \begin{equation*} \bigoplus_{n\geq 0}\operatorname{HH}^*(\operatorname{Hilb}^nS)t^n \cong \operatorname{Sym}^\bullet(\bigoplus_{i\geq 1}\operatorname{HS}_{1-i}^*(S)t^i) \end{equation*}

Conclusion

Sure, this isn't good enough to write down the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of the Hilbert scheme, but at least we can write down the Hochschild cohomology. And in fact, for the pieces we care about the most (namely the first and second Hochschild cohomology) we know enough to write down the Hochschild–Kostant–Rosenberg decomposition!

In fact, we also:

  • work it all out for arbitrary symmetric quotient stacks
  • have versions of the result where we allow coefficients
  • explain various examples from different angles, and discuss applications to the deformation theory of Hilbert schemes
  • discuss Hochschild–Serre cohomology for dg categories, as a Morita invariant
  • explain étale functoriality for Hochschild–Serre cohomology
so there is plenty of stuff I haven't discussed in this blogpost.

I am really proud of this paper, and I'd be happy to hear all comments, and answer any questions you might have!