Problem 24.4.31EF3C6D763307

Problem 24.4.31.EF3C6D763307 In the setting of (24.4.30), find geometric hypotheses that control the integral on the right-hand side, the cost of the subtraction, up to a nef endpoint, beyond the regime of a closed nef core treated in theorem 12.12.2. Determine when positive root volume forces bigness in the absence of such a core. For closed backgrounds, compare the resulting estimate with the classical Morse inequalities; for nonclosed backgrounds, the estimate should keep track of the differential data and of the dependence on $\mu $.