11.3 Gauduchon metrics and linear mass1009E65ECC4C

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Throughout this chapter, $\omega $ is a fixed Hermitian form on $X$, and $n\geq 2$ unless a section specifies another setting.

The mass variation in section 11.1 leaves a linear conservation law. If $\gamma $ is a Gauduchon metric, so that $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$, integration against $\gamma ^{n-1}$ annihilates every complex Hessian. Thus the trace mass $\int _ X(\theta +\mathrm{dd}^{\mathrm{c}}u)\wedge \gamma ^{n-1}$ of a positive current depends only on its smooth background $\theta $, even when $u$ is unbounded. Gauduchon's theorem produces such a metric in every conformal class, providing conserved pairings when the background is not closed.

These pairings determine positivity as well as mass. A smooth form $\theta $ admits a positive current $\theta +\mathrm{dd}^{\mathrm{c}}u$ exactly when all its Gauduchon degrees are nonnegative. Normalizing the degrees gives the largest lower curvature bound obtainable by adding a complex Hessian. This bound is always attained by a potential; a minimizing test exists in the weak closure of the normalized smooth Gauduchon forms, although a smooth minimizing metric generally does not exist. Applied to $\omega _ u$, the conformal construction also gives the adjoint probability of the linearized Monge–Ampère operator. It identifies the compatibility condition for that operator and explains the extra term in entropy dissipation for a nonclosed background.

For a holomorphic line bundle, the conserved pairing is the degree of its curvature, and a positive degree excludes a metric with nonpositive curvature. Fiber averaging extends this obstruction to plurisubharmonic norms near a direct sum of line bundles. Polar duality and homogeneous regularization then turn it into an obstruction to semipositive Finsler curvature. The stable ample bundles constructed on Hirzebruch surfaces show that ampleness alone does not remove the obstruction; their projective complements give a geometric way to measure it.

The same scalar equation can improve curvature when the mean curvature is nonnegative: a conformal change spreads directional positivity from a nonempty open set to the whole manifold. Ricci-flat Kähler surfaces illustrate the distinction between this directional condition and positivity on every tangent and fiber pair. On a projective manifold, positivity on every such pair admits another duality criterion, expressed by positive measures over the dual bundle.