Monge-Amp\`ere-type equation for forms of positive degree and Demailly's transcendental Morse inequality

Abstract

The Monge-Amp\`ere-type equation for forms of positive degree was introduced by Dinew and Popovici to prove the qualitative part of an analytic version of Demailly's transcendental Morse inequality for higher cohomology classes, conditional on the solvability of this nonlinear PDE. In this paper, we show that this can be proven unconditionally. We first show that the originally proposed Laplacian trace condition, $\Lambda^{m-2}\Delta u = 0$, is analytically too rigid to permit solutions and hence introduce a general gauge-fixing that leads to the $(a,b)$ Monge-Amp\`ere-type equation. By deriving \emph{a priori} estimates, we establish the solvability of this class of equations for different parametric values $(a,b)$. As a geometric application, for $b=0$ this framework reduces to the classical complex Monge-Amp\`ere equation, which yields the qualitative Demailly's inequality for higher-degree forms. Further, we prove that the uniqueness of these solutions can be significantly strengthened under the kernel condition $\bar{\partial}^* u = 0$.