Fu-Yau Hessian Equations

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope ${\alpha }^{\mathrm{\prime }}$. Actually we obtain at the same time a solution of the open case ${\alpha }^{\mathrm{\prime }}>0$, an improved solution of the known case ${\alpha }^{\mathrm{\prime }}<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual ${\mathrm{\Gamma }}_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.