Fourier–Mukai/decomposition correspondence for integrable systems

Let $\pi_M:(M,\sigma)\to B$ be an integrable system, with partial Fourier–Mukai transform $\phi_{\mathrm{FM}}$, sheaf of Kähler differentials $\Omega_M^k$, and decomposition-theorem Hodge module $P_k$ whose associated graded object is $\operatorname{gr}(P_k)\in D^b\operatorname{Coh}(T^*B)$. Using a relatively ample line bundle to identify the formal neighborhoods of the zero section in $P$ and $T^*B$, write this neighborhood as $\hat{B}$. Fourier–Mukai/decomposition correspondence. One has

$$\phi_{\mathrm{FM}}(\Omega_M^k)|_{\hat{B}}\simeq \operatorname{gr}(P_k)|_{\hat{B}}\in D^b\operatorname{Coh}(\hat{B}).$$

The correspondence proposes that the Fourier–Mukai transforms of differential forms near the zero section are quantized by the Hodge modules arising from the decomposition theorem. Its precise formulation depends on the formal-neighborhood identification and is presented as the paper’s main conjectural relationship.