Volume and Self-Intersection of Differences of Two Nef Classes

Let $\{\alpha\}$ and $\{\beta\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional K\"ahler manifold $X$ such that the difference of intersection numbers $\{\alpha\}^n - n\,\{\alpha\}^{n-1}.\,\{\beta\}$ is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transcendental Morse Inequalities Conjecture for this context predicting the lower bound $\{\alpha\}^n-n\,\{\alpha\}^{n-1}.\,\{\beta\}$ for the volume of the difference class $\{\alpha-\beta\}$. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of $\{\alpha-\beta\}$ and show that the self-intersection number $\{\alpha-\beta\}^n$ is always bounded below by $\{\alpha\}^n-n\,\{\alpha\}^{n-1}.\,\{\beta\}$. We also describe and estimate the relative psef and nef thresholds of $\{\alpha\}$ with respect to $\{\beta\}$ and relate them to the volume of $\{\alpha-\beta\}$. Finally, broadening the scope beyond the K\"ahler realm, we propose a conjecture relating the balanced and the Gauduchon cones of $\partial\bar\partial$-manifolds which, if proved to hold true, would imply the existence of a balanced metric on any $\partial\bar\partial$-manifold.