On the modified $J$-equation

In this paper, we study the modified $J$-equation introduced by Li-Shi. We first show that, on compact K\"ahler manifolds, the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional. Motivated by this characterization, we formulate a Nakai-Moishezon type criterion for the existence of solutions to the modified $J$-equation on general compact K\"ahler manifolds. We then verify this conjectural criterion in the case of smooth projective toric varieties. This extends the work of Collins-Sz\'ekelyhidi and provides further evidence for the expected algebro-geometric nature of the modified $J$-equation. As a potential application, we combine our results with Delcroix-Jubert. Assuming our conjectural Nakai-Moishezon type criterion holds in general, we obtain a numerical sufficient condition for the existence of extremal K\"ahler metrics on arbitrary compact K\"ahler manifolds.