24.1 Problems in Hermitian pluripotential theory9226B45701D5

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Throughout this chapter $X$ is a connected compact complex manifold of dimension $n\geq 1$. The Hermitian forms, volume forms and backgrounds needed for each problem are fixed in the section where the problem is stated.

Many arguments here begin with linear control. A uniform bound on Gauduchon traces makes a family of positive currents relatively compact, and Gauduchon duality expresses scalar invariants through optimization over test currents. Most of the questions below arise because such linear control does not determine the nonlinear data of a limit, namely its Monge–Ampère mass and its singularities. Let $\omega $ be a Hermitian form on $X$ that fails the bounded mass property of definition 13.14.14. By proposition 13.14.21, corollary 13.14.22, currents $\omega +\mathrm{dd}^{\mathrm{c}}u$ with integrable coefficients can converge in the coefficient $L^1$ norm, with Gauduchon trace measures converging in total variation, while their determinant masses are infinite. By proposition 18.5.5, even the relative envelope class $\mathcal E_{\mathrm{env}}(X,\omega ,0)$ of definition 16.28.61 contains potentials of infinite mass. Passing from linear bounds to an exact equation therefore requires additional control of the nonlinear limit.

Two further requirements recur. Positivity of a scalar invariant does not imply strict positivity: the smooth semipositive background on a projective manifold in theorem 12.14.6 has positive Gauduchon variational volume but is not big. Likewise, an envelope realizing prescribed divisorial lower bounds may acquire larger orders, and then it does not recover the prescribed singularity type. Thus the mass of a limiting solution, strict positivity of a current, and exactness of its divisorial orders have to be checked in their respective arguments.