Pome

Proposition 12.16.2CE7BE3992C5F

Proposition 12.16.2.CE7BE3992C5F Let $n\geq 3$, and let $v\in \mathcal H_\omega $ be a local minimum of $\mathrm M_\omega $ for the $C^2$ topology. Then $\Omega =\omega +\mathrm{dd}^{\mathrm{c}}v$ satisfies

\[ \mathrm{dd}^{\mathrm{c}}(\Omega ^{n-1})=0, \qquad \mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})\leq 0. \]

At a local maximum the first condition still holds and the second inequality is reversed.

Proof. For every smooth real $f$, the potentials $v+tf$ lie in $\mathcal H_\omega $ for all sufficiently small $|t|$. The vanishing of the first variation therefore gives the Gauduchon condition. The second variation is

\[ \begin{aligned} \left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}\right|_{t=0}\mathrm M_\omega (v+tf) & =n(n-1)\int _ X(\mathrm{dd}^{\mathrm{c}}f)^2\wedge \Omega ^{n-2}\\ & =-n(n-1)\int _ X\mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge \mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2}), \end{aligned} \]

where the last equality follows by integrating by parts twice, using that $\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})$ is both $\mathrm{d}$-closed and $\mathrm d^{\mathrm c}$-closed. At a local minimum this second variation is nonnegative for every $f$, and lemma 12.16.1, applied to $-\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})$, gives $\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})\leq 0$. The case of a maximum is identical, with the opposite sign. $\square$

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