PomeTheorem 24.4.20.781DAEC1B4CE Let $D$ be a connection on $TX$ as above, preserving $g$ and $J$ and with parallel torsion. If $X$ admits a balanced metric and a pluriclosed metric, then it admits a $D$-parallel Kähler metric.
Theorem 24.4.20781DAEC1B4CE
Proof. Let $\beta $ be a balanced metric and $\omega $ a pluriclosed metric. By lemma 24.4.17, $S(\beta ^{n-1})$ is strictly positive, parallel and closed. It has a unique smooth positive $(n-1)$-st root $B$. Indeed, after the fixed normalization of wedge powers, the corresponding map on matrices is $G\mapsto C=(\det G)G^{-1}$, whose inverse is
\[ G=(\det C)^{1/(n-1)}C^{-1}. \]
By uniqueness the root is invariant under parallel transport. Thus $DB=0$ and $\mathrm dB^{n-1}=0$. Similarly, $P=S\omega $ is a parallel positive pluriclosed form.
Consider the finite-dimensional space of real parallel directions of exact type
\[ \mathcal A_ D=\{ (\mathrm d\xi )^{1,1}:\xi \in E_ D^1\} . \]
Every such direction is $\partial \bar\partial $-closed, since
\[ (\mathrm d\xi )^{1,1}=\partial \xi ^{0,1}+\bar\partial \xi ^{1,0}. \]
For a parallel positive form $Q$, the volume functional has derivative
B755FCE3BC15
24.4.21
\begin{equation} \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} \frac1{n!}\int _ X(Q+t(\mathrm d\xi )^{1,1})^ n =\frac1{(n-1)!}\int _ X\xi \wedge \mathrm dQ^{n-1}, \end{equation}
because the $(2,0)$ and $(0,2)$ parts of $\mathrm d\xi $ pair to zero with $Q^{n-1}$. Since $\mathrm dQ^{n-1}$ is parallel, nondegeneracy of the parallel pairing shows that all these derivatives vanish exactly when $Q$ is balanced.
Apply this to the balanced form $B$. For $a\in \mathcal A_ D$, the function $\operatorname {tr}_ B a$ is parallel and hence constant. Its integral against $B^ n$ vanishes by (24.4.21), so it vanishes identically. Therefore every form in the positive affine slice
\[ \mathcal P=\{ Q\in P+\mathcal A_ D:Q>0\} \]
has the same pointwise trace $\operatorname {tr}_ B Q=\operatorname {tr}_ B P=:C>0$.
Evaluate the parallel forms at one point. The positive eigenvalues of $Q\in \mathcal P$ relative to $B$ sum to $C$, so they are bounded above by $C$. Thus $\mathcal P$ is bounded in its finite-dimensional affine space. Its closure is
\[ \overline{\mathcal P}=\{ Q\in P+\mathcal A_ D:Q\geq 0\} , \]
because $(1-t)Q+tP$ is positive for $0<t\leq 1$ whenever $Q$ is semipositive; in particular, this closure is compact. Since both forms are parallel, the relative eigenvalues and the relative determinant are constant on $X$. Hence
\[ \frac1{n!}\int _ XQ^ n =\det (B^{-1}Q)\frac1{n!}\int _ XB^ n. \]
At a relative boundary point of $\overline{\mathcal P}$ the semipositive matrix is singular, so this volume is zero, whereas it is positive at $P$. A maximum of the volume on the compact set $\overline{\mathcal P}$ therefore lies in $\mathcal P$.
At this interior maximum all the derivatives in (24.4.21) vanish, so the maximizing form is balanced; it is pluriclosed because it lies in $P+\mathcal A_ D$. By lemma 24.4.19 it is closed. Since it is positive and $D$-parallel, this proves the theorem. $\square$
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