Fourier–Mukai compatibility conjecture with explicit complexified ample classes

Let $X$ and $Y$ be abelian varieties of dimension $g$, let $D_X$ and $D_Y$ be the associated classes, let $\ell_X$ and $\ell_Y$ be ample classes, and let $\Phi_{\mathcal{E}}^{X\to Y}:D^b(X)\to D^b(Y)$ be the Fourier–Mukai transform. For $k\in\{1,2,\ldots,g-1\}$ and $\lambda\in\mathbb{R}_{>0}$, define

$$\Omega_k=-D_X+\lambda e^{ik\pi/g}\ell_X, \qquad \Omega'_k=D_Y-e^{-ik\pi/g}\ell_Y/\lambda.$$

Let $\mathcal{A}^{X}_{\Omega_k}$ and $\mathcal{A}^{Y}_{\Omega'_k}$ be the conjecturally constructed iterated-tilt hearts.

Fourier–Mukai compatibility conjecture. One has

$$\Phi_{\mathcal{E}}^{X\to Y}[k](\mathcal{A}^{X}_{\Omega_k})=\mathcal{A}^{Y}_{\Omega'_k}.$$

This is the explicit real-phase specialization of the Fourier–Mukai compatibility expectation, with the classes determined by the Fourier–Mukai action on the central charge. It remains conditional on the existence of the relevant stability conditions and is open in general.