Polyvectors and Hochschild cohomology
- Polyvector parallelogram
- Each entry is $\mathrm{h}^q(X,\bigwedge^p\mathrm{T}_X)$. Row $k$ has $p+q=k$, with $p$ increasing from left to right. The Hochschild–Kostant–Rosenberg decomposition identifies its row sum with $$\begin{equation} \notag \dim\mathrm{HH}^k(X)=\sum_{p+q=k}\mathrm{h}^q(X,\bigwedge^p\mathrm{T}_X). \end{equation}$$
- Vanishing entries
- Akizuki–Nakano vanishing gives zero when $q>p$; Kodaira vanishing also gives $\mathrm{H}^q(X,\omega_X^\vee)=0$ for $q>0$.
- Dependence on the model
- These dimensions can jump in a deformation family, so the displayed values belong to the selected presentation. For the general definitions, see Polyvector fields for Fano 3-folds, §2.