24.4.29 Morse inequalities and the choice of volumeB20929459DA6

In the closed Kähler theory, subtracting a nef class has a controlled cost in volume. For closed nef classes $\alpha ,\beta $ on a compact Kähler manifold, the transcendental Morse inequality states that

where $\operatorname {Vol}$ is the usual volume of a class. Its source, and the differentiability of the volume and the duality of cones that follow from it, are recalled in section 25.22. To formulate an analogue on an arbitrary compact complex manifold, one must first choose the volume to be estimated.

For nonclosed backgrounds, the intersection numbers on the right-hand side are in general not invariant under a change of representatives. Moreover, the upper volume of a class and $\mathcal V_\mu $ have different definitions and can behave differently with respect to finiteness. A Hermitian analogue must therefore specify which invariant it estimates.

The first-variation formula gives a finite quantity to study. Fix smooth real $(1,1)$-forms $\theta ,\eta $ and $t>0$. Suppose that $\theta _ s=\theta -s\eta $ has a smooth positive representative for $0\leq s\leq t$, and let $\gamma _ s$ be a minimizing Gauduchon metric for its smooth equation. By proposition 12.3.2 and the fundamental theorem of calculus,

The quotient in the integrand does not change when the minimizer is rescaled. This identity holds on the positive segment $0\leq s\leq t$ and uses no bound on the upper Monge–Ampère mass. A uniform upper bound for its integrand gives a lower bound for the remaining root volume. On a compact Kähler manifold, theorem 12.12.2 extends the identity up to a nef endpoint that has no smooth positive representative, and bounds the resulting improper integral, when a closed nef core of positive volume is supplemented by a semipositive form $B$. Its test metrics keep the prescribed density, and (12.12.5) isolates the correction involving $\mathrm{dd}^{\mathrm{c}}B$. There the endpoint is big because of the closed core; example 12.12.7 gives a nonclosed example with an explicit strict current at the endpoint. Positive volume at the endpoint alone cannot provide such a current, by theorem 12.14.6, so a criterion going beyond the closed core must exclude that obstruction.

For the warped products of section 12.13, theorem 12.13.1 computes the finite improper cost for every choice of minimizing metric, and proposition 12.13.4 gives its rate from the codimension and vanishing order of the fiber coefficient. These paths can end with positive root volume but no big representative, and proposition 12.13.6 excludes every closed nef positive-volume core at a zero of that coefficient. Thus exact control of the subtraction cost in this product family does not supply a general bigness criterion.

The singularity and divisorial constructions of these constructions become relevant here: they measure how much positivity can be spent on imposed poles while the residual Gauduchon volume stays finite. They do not yet give the moving intersection product or the birational volume formula of the closed theory. For the nonanalytic ruled type of example 12.35.13, the improper subtraction cost remains finite as the variational volume vanishes, while the upper ordinary-type volume has a positive limit. This family makes the choice of invariant explicit even for an imposed type with infinite first moment. Recovering a general moving product or birational formula remains a further goal.