Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds (Q2795675)

The author considers the Monge-Ampère type equation of bidegree \((n-1,n-1)\) on a compact complex manifold with \(\dim_{\mathbb C}X=n>2\). Let \(\omega\) be a Gauduchon metric on \(X\) which means that \(\partial\overline{\partial}\omega^{n-1}=0\). Then the author considers the equation NEWLINE\[CARRIAGE_RETURNNEWLINE[ (\omega^{n-1}+i\partial\overline{\partial}\phi\wedge \omega^{n-2}+\frac i2(\partial\phi\wedge \overline{\partial}\omega^{n-2}-\overline{\partial}\phi\wedge \partial\omega^{n-2}))^{\frac1{n-1}}]^n=e^f\omega^n\eqno{(\ast)}CARRIAGE_RETURNNEWLINE\]NEWLINE subject to the positivity and normalization conditions NEWLINE\[CARRIAGE_RETURNNEWLINE\omega^{n-1}+i\partial\overline{\partial}\phi\wedge \omega^{n-2}+\frac i2(\partial\phi\wedge \overline{\partial}\omega^{n-2}-\overline{\partial}\phi\wedge \partial\omega^{n-2})>0CARRIAGE_RETURNNEWLINE\]NEWLINE and \(\sup_X\phi=0\) where for a \((n-1,n-1)\)-form \(\Gamma\), the form \(\Gamma^{\frac{1}{n-1}}\) is the unique \((1,1)\)-form \(\gamma\) such that \(\Gamma=\gamma^{n-1}\). The author proves that the solution \(\phi\) to \((\ast)\) under the positivity and normalization conditions is unique. The author also proves that the principal part of the linearization of Equation \((\ast)\) is the Laplacian associated with a certain Hermitian metric on \(X\). The investigation of \((\ast)\) is associated with the question to which extent there is an Aeppli-Gauduchon analogue of Yau's theorem of the Calabi conjecture. In the paper the author also studies the Bott-Chern and Aeppli cohomologies in particular on \(\partial\overline{\partial}\) complex manifolds. He also introduces the Gauduchon and sG cones in \(H_A^{n-1,n-1}(X,\mathbb R)\) and studies their properties.