Volume and Self-Intersection of Differences of Two Nef Classes

Abstract

Let $\{\alpha \}$ and $\{\beta \}$ be nef cohomology classes of bidegree $(1,1)$ on a compact $n$-dimensional Kähler 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 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.

40 pages

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