Pome

17.16.8 The smooth product equationA928C3A148CF

For a smooth path, positivity on the product compares the acceleration with the spatial gradient of the velocity. Let $u_ t$ be smooth in $(x,t)\in X\times I$, suppose that $\eta _ t=\theta +\mathrm{dd}^{\mathrm{c}}u_ t>0$, and write, in local holomorphic coordinates $(x_1,\ldots ,x_ n)$,

\[ \theta =\frac{\mathrm i}{2\pi }\sum _{i,j=1}^ n g_{i\bar j}\, \mathrm dx_ i\wedge \mathrm d\bar x_ j, \qquad H=(g_{i\bar j}+\partial _ i\partial _{\bar j}u), \qquad \xi =(\partial _ i\dot u)_{i=1}^ n. \]

Here dots denote derivatives in $t$, $\xi $ is a column vector, and $\xi ^*$ is its conjugate transpose. Since $t=\operatorname{Re}z$, the form $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}U$ on the product has, in the coordinates $(x_1,\ldots ,x_ n,z)$, the coefficient matrix

\[ \begin{pmatrix} H & \tfrac 12\xi \\ \tfrac 12\xi ^* & \tfrac 14\ddot u \end{pmatrix}. \]

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). $\square$

This calculation identifies the homogeneous equation for smooth paths. Whether nonsmooth Perron segments solve a weak form of this equation is a separate question, which we do not address.

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