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11.29.2 The finite-dimensional alternative6243CAB7B202

For $m\geq m_0$, choose a basis of $V_ m=H^0(X,E\otimes A^ m)$ and write $N_ m=\dim V_ m$. Pairing these sections with $v\in F_ x$ and using a local frame of $A^ m$, we obtain a column $p_ m(v)$ of local holomorphic functions on the total space of $F$. For any Hermitian $N_ m\times N_ m$ matrix $B$, set

\[ G_{m,B}(v)=\mathrm e^{-m\phi (x)}p_ m(v)^*B p_ m(v). \]

This function is globally defined and quadratic on each fiber, so $G_{m,B}=G_{h_{m,B}}$ for a smooth Hermitian form $h_{m,B}$ on $F$. The matrix $B$ is allowed to have negative eigenvalues.

For $(v,\xi )\in K$, let $u$ be the projection of $\xi $ to $T_ x^{1,0}X$, write $\phi _ u=\partial \phi (u)$, and put

\begin{align*} r_ m& =\mathrm dp_ m(\xi )-m\phi _ u p_ m(v),\qquad \kappa =\phi _{u\bar u},\\ \Psi _ m(v,\xi )& =\mathrm e^{-m\phi } \bigl (r_ mr_ m^*-m\kappa p_ m p_ m^*\bigr ). \end{align*}

Differentiating the weighted quadratic function gives

F67D74886176 11.29.3
\begin{equation} (D h_{m,B})_ v(\xi ,\xi )=\operatorname {tr}(B\Psi _ m(v,\xi )). \end{equation}

All entries involve only the first jets of the sections: if $t=(t_\alpha )$ is the column of the sections of $A$, then

\[ \phi _ u=\frac{t^*\mathrm dt(u)}{\rho },\qquad \kappa =\frac{\rho |\mathrm dt(u)|^2-|t^*\mathrm dt(u)|^2}{\rho ^2}. \]

Changing the frame of $A$ by a nowhere-zero holomorphic factor $\lambda $ changes $p_ m$ to $\lambda ^{-m}p_ m$ and $\phi $ to $\phi -\log |\lambda |^2$. It changes $r_ m$ by the same factor and leaves $\kappa $ unchanged, so $\Psi _ m$ is frame independent. A change of the basis of $V_ m$ replaces $\Psi _ m$ by a congruent matrix, so whether the convex hull of the values of $\Psi _ m$ contains zero does not depend on that choice.

Proposition 11.29.4.07EC0797039C At each fixed $m$, exactly one of the following holds:

  1. Some Hermitian matrix $B$ satisfies $\operatorname {tr}(B\Psi _ m)>0$ everywhere on $K$.

  2. There is a probability measure $\mu $ on $K$, supported on at most $N_ m^2+1$ points, such that $\int _ K\Psi _ m\, \mathrm d\mu =0$.

The first alternative holds precisely when $0\notin \operatorname {conv}(\Psi _ m(K))$.

Proof. The real vector space of Hermitian $N_ m\times N_ m$ matrices has dimension $N_ m^2$. Carathéodory's theorem expresses every point of the convex hull as a convex combination of at most $N_ m^2+1$ points of the compact image $\Psi _ m(K)$, so the hull is compact. If it contains zero, the coefficients of such an expression define the required atomic probability measure $\mu $, and then $\int _ K\operatorname {tr}(B\Psi _ m)\, \mathrm d\mu =0$ excludes the first alternative.

If the hull misses zero, let $Z_0$ be its point of least Hilbert–Schmidt norm. Differentiating the squared norm along the segment from $Z_0$ to any $Z$ in the hull gives

\[ \operatorname {tr}(Z_0Z)\geq \operatorname {tr}(Z_0^2)>0. \]

Taking $B=Z_0$ proves the first alternative. $\square$

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