Closure of the Calabi continuity path under uniform estimates

Use the fixed chart balls $D_{\nu }$ from the Statement. For every multi-index $\beta$ of order at most $r$, the uniform $C^{r+1}({\omega }_0)$ bound makes ${\mathrm{\partial }}^{\beta }({\varphi }_j\circ {\kappa }_{\nu }^{-1})$ uniformly bounded and equicontinuous on each closed ball: the mean-value inequality bounds its oscillation by a constant times the Euclidean distance, using the uniform bounds on derivatives of order $\mathrm{\mid }\beta \mathrm{\mid }+1$. Apply the Euclidean Arzelà–Ascoli theorem [F2] to the finite collection of derivative arrays and take a finite diagonal subsequence. Uniform convergence of all derivatives through order $r$ gives convergence in the stated $C^r({\omega }_0)$ norm. Successively do this for $r=0,1,2,\dots$ and take the diagonal subsequence. On each chart interior the derivative limits are the derivatives of the limiting function, by the fundamental theorem of calculus on coordinate line segments; the limits agree on overlaps because each ${\varphi }_j$ does. They therefore define a smooth real ${\varphi }_{\mathrm{\infty }}$, and the subsequence converges in every stated $C^r$ norm.

Convergence in $C^0$ preserves ${\int }_X{\varphi }_{\mathrm{\infty }}{\omega }_0^n=0$. Convergence in $C^2$ gives ${\omega }_{\mathrm{\infty }}={\omega }_0+i\mathrm{\partial }\bar{\mathrm{\partial }}{\varphi }_{\mathrm{\infty }}$ as the limit of positive forms, so ${\omega }_{\mathrm{\infty }}$ is semipositive. Smooth dependence of ${\mu }_t$ on $t$ and $C^2$ convergence let the determinant equation pass to ${\omega }_{\mathrm{\infty }}^n={\mu }_{t_{\mathrm{\infty }}}$; the right side is strictly positive. At each point the eigenvalues of ${\omega }_{\mathrm{\infty }}$ relative to ${\omega }_0$ are nonnegative and their product is positive, so each is positive. Thus ${\omega }_{\mathrm{\infty }}$ is Kähler and the limit has every required property.