Pome

Corollary 11.2.2167DF1C6C0509

Corollary 11.2.21.67DF1C6C0509 For a Hermitian symplectic form $\omega $ on $X$, the following conditions are equivalent:

  1. $F(\omega +\mathrm{dd}^{\mathrm{c}}\varphi )$ is independent of the smooth real function $\varphi $ with $\omega +\mathrm{dd}^{\mathrm{c}}\varphi >0$;

  2. $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$;

  3. $\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^4$ is independent of the bounded $\omega $-plurisubharmonic function $u$.

The common mass in the third condition is $24V(\omega )$. Every local minimum of $F$ in an Aeppli class has these properties.

Proof. By (11.2.18), the energy is constant along smooth positive potential variations if and only if their volume is. Since $\mathrm{dd}^{\mathrm{c}}\omega =0$, lemma 11.1.3, theorem 11.1.7 show that this smooth mass invariance is equivalent to $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$ and to mass invariance for bounded potentials. The zero potential gives the common mass. The last assertion follows from theorem 11.2.19. $\square$

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