14.10 Prescribing a Gauduchon volume form6950E7B1FBCB

We now return to a general connected compact complex manifold $X$, of dimension $n\geq 2$. To stay in a fixed Aeppli class of Gauduchon metrics, we vary the $(n-1)$st power of the metric by a term of the form $\partial \xi +\bar\partial \bar\xi $. The resulting scalar ansatz contains first derivatives of the potential as well as its complex Hessian.

Here we use the linear-algebra fact that a real $(n-1,n-1)$-form is strictly positive if and only if it is the $(n-1)$st power of a Hermitian form, and that this Hermitian form is unique; the matrix form of this correspondence is described in section 14.13. Keep $\gamma _0$ and $\gamma $ fixed for the remainder of the chapter, and write $\mathcal D_\gamma u=\Gamma _ u-\gamma _0^{n-1}$ for the part of the ansatz that is linear in $u$.

To see which geometry is preserved, set $\xi _ u=\mathrm i\, \bar\partial u\wedge \gamma ^{n-2}/(4\pi )$. The Leibniz rule gives $\Gamma _ u-\gamma _0^{n-1}=\partial \xi _ u+\bar\partial \overline{\xi _ u}$. Since $\mathrm{dd}^{\mathrm{c}}(\partial \xi _ u+\bar\partial \overline{\xi _ u})=0$, if $\gamma _0$ is Gauduchon then so is $\eta _ u$, and the $(n-1)$st powers of $\gamma _0$ and $\eta _ u$ have the same Aeppli class. The uniqueness in theorem 14.10.1 concerns this scalar ansatz; it does not assert uniqueness among all Gauduchon metrics in the Aeppli class.