Proposition 17.16.9.5885FD8F9550 The form $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}U$ is semipositive exactly when
\[ \ddot u\geq \xi ^*H^{-1}\xi . \]
Its $(n+1)$st power vanishes exactly when equality holds. Equivalently, the homogeneous Monge–Ampère equation on the product is
F12E6A9111B5
17.16.10
\begin{equation} \ddot u\, \eta _ t^ n =n\, \mathrm{d}\dot u\wedge \mathrm d^{\mathrm c}\dot u\wedge \eta _ t^{n-1}. \end{equation}
Proof.
The Schur complement of $H$ in the block matrix is $(\ddot u-\xi ^*H^{-1}\xi )/4$. Since $H>0$, the block matrix is semipositive exactly when this complement is nonnegative, and its determinant, which is $\det H$ times the complement, vanishes exactly when the complement vanishes.
To rewrite the equation in terms of forms, note that
\[ \mathrm{d}\dot u\wedge \mathrm d^{\mathrm c}\dot u=\frac{\mathrm i}{2\pi }\partial \dot u\wedge \bar\partial \dot u, \qquad n\frac{\mathrm{d}\dot u\wedge \mathrm d^{\mathrm c}\dot u\wedge \eta _ t^{n-1}}{\eta _ t^ n} =\operatorname {tr}(H^{-1}\xi \xi ^*)=\xi ^*H^{-1}\xi . \]
Multiplying the second identity by $\eta _ t^ n>0$ gives (17.16.10).
$\square$
Pome
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