Ricci-flat metric in every Kähler class when c₁=0

Fix a Kähler class and a representative ${\omega }_0$. The Ricci form ${\rho }_0$ is real, closed, and represents $2\pi c_1(T_X{)}_{\mathbb{R}}=0$ by [F2], so it is $d$-exact. It is a pure-type $(1,1)$-form; under the full AC hypothesis [A1], [F3] gives a smooth complex function $\beta$ with ${\rho }_0=\mathrm{\partial }\bar{\mathrm{\partial }}\beta$.

Let ${\omega }_1,{\omega }_2$ be Ricci-flat Kähler forms in the same class. Their difference is real, $d$-closed, $d$-exact, and of type $(1,1)$; under [A1], [F3] writes ${\omega }_1-{\omega }_2=\mathrm{\partial }\bar{\mathrm{\partial }}\gamma$ for a smooth complex function $\gamma$. The reality relation implies $\psi =(\gamma -\bar{\gamma })\mathrm{/}(2i)$ is real and ${\omega }_1-{\omega }_2=i\mathrm{\partial }\bar{\mathrm{\partial }}\psi$.

Since ${\rho }_0$ is real, complex conjugation and $\overline{\mathrm{\partial }\stackrel{ˉ}{\mathrm{\partial }}\beta }=-\mathrm{\partial }\bar{\mathrm{\partial }}\bar{\beta }$ give $\mathrm{\partial }\bar{\mathrm{\partial }}\beta =-\mathrm{\partial }\bar{\mathrm{\partial }}\bar{\beta }$. Hence the real function $F=(\beta -\bar{\beta })\mathrm{/}(2i)$ satisfies $i\mathrm{\partial }\bar{\mathrm{\partial }}F={\rho }_0$. Add the real constant $c=\log (V\mathrm{/}{\int }_X{e}^F{\omega }_0^n)$, where $V={\int }_X{\omega }_0^n$, so that ${\int }_X{e}^{F+c}{\omega }_0^n=V$.

The Ricci variation formula in [F2] gives $\mathrm{\partial }\bar{\mathrm{\partial }}\log ({\omega }_1^n\mathrm{/}{\omega }_2^n)=0$. Its trace is the scalar Laplace equation, so the real logarithm is harmonic. On compact connected $X$, integration by parts [F5] gives ${\int }_X\mathrm{\mid }d\log ({\omega }_1^n\mathrm{/}{\omega }_2^n){\mathrm{\mid }}^{2}dV=0$; hence the ratio is constant. Since [step 1.2] places the two forms in the same class, their top-volume integrals agree by Stokes [F5], the constant is $1$, and ${\omega }_1^n={\omega }_2^n$.

Set $\mu =e^{F+c}{\omega }_0^n$, a smooth positive volume form with the required total volume. Apply the theorem in [F1] to obtain a normalized real $\varphi$ and ${\omega }_{\varphi }={\omega }_0+i\mathrm{\partial }\bar{\mathrm{\partial }}\varphi >0$ with ${\omega }_{\varphi }^n=\mu$.

In the fixed holomorphic coordinates of [F2], determinant multiplication gives $\rho ({\omega }_{\varphi })=\rho ({\omega }_0)-i\mathrm{\partial }\bar{\mathrm{\partial }}\log ({\omega }_{\varphi }^n\mathrm{/}{\omega }_0^n)$. The ratio is $e^{F+c}$, so the right side is $i\mathrm{\partial }\bar{\mathrm{\partial }}F-i\mathrm{\partial }\bar{\mathrm{\partial }}(F+c)=0$. Thus the constructed form is Ricci-flat. Its cohomology class is the chosen class because $i\mathrm{\partial }\bar{\mathrm{\partial }}\varphi$ is exact.

If $K_X$ is holomorphically trivial, [F4] gives the real-class hypothesis, so the same construction applies to each Kähler class.

Put $S={\sum }_{j=0}^{n-1}{\omega }_1^j\wedge {\omega }_2^{n-1-j}$, which is closed and positive. The difference-of-powers identity, the real potential in [step 1.2], and the equal volume forms in [step 2.2] give $0={\int }_X\psi ({\omega }_1^n-{\omega }_2^n)={\int }_X\psi i\mathrm{\partial }\bar{\mathrm{\partial }}\psi \wedge S=-{\int }_{X}i\mathrm{\partial }\psi \wedge \bar{\mathrm{\partial }}\psi \wedge S$ by [F5]. The last integrand is nonnegative and vanishes only where $d\psi =0$, so $\psi$ is constant on connected $X$ and ${\omega }_1={\omega }_2$. This proves uniqueness independently of the existence estimates.