Volume and Self-Intersection of Differences of Two Nef Classes

Abstract

Let $\{α\}$ and $\{β\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional Kähler manifold $X$ such that the difference of intersection numbers $\{α\}^n - n\,\{α\}^{n-1}.\,\{β\}$ 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 $\{α\}^n-n\,\{α\}^{n-1}.\,\{β\}$ for the volume of the difference class $\{α-β\}$. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of $\{α-β\}$ and show that the self-intersection number $\{α-β\}^n$ is always bounded below by $\{α\}^n-n\,\{α\}^{n-1}.\,\{β\}$. We also describe and estimate the relative psef and nef thresholds of $\{α\}$ with respect to $\{β\}$ and relate them to the volume of $\{α-β\}$. Finally, broadening the scope beyond the Kähler 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.