# What are the known examples of compact complex $n$-folds that do not have a strongly Gauduchon metric for $n >2$?

Gauduchon showed that every conformal hermitian structure on a compact complex $n$-fold contains an hermitian metric such that the associated 1,1-form $\omega$ satisfies $\partial {\bar{\partial}}\omega^{n-1} = 0$. Such metrics are called Gauduchon metrics. Popovici defines a strongly Gauduchon metric as a Gauduchon metric such that $\partial \omega^{n-1}$ is ${\bar{\partial}}$-exact. For $n>2$, what are the known examples of compact complex $n$-folds that do not admit a strongly Gauduchon metric?

Probably the simplest example is the following: $X$ is the Hopf manifold $({\mathbb C}^3\setminus\{0\})/<\gamma>$ where $\gamma(z_1,z_2,z_3)=(2z_1,2z_2,2z_3)$. It has a projection $p:X\to {\mathbb P}^2$, $p(z_1,z_2,z_3)=[z_1:z_2:z_3]$. It is diffeomorphic to $S^5\times S^1$, hence $H^2(X, {\mathbb R})=0$. If $\omega_{{\mathbb P}^2}$ denotes the Fubini-Study metric on ${\mathbb P}^2$, then $p^*\omega_{{\mathbb P}^2}$ is a closed, positive $(1,1)$-form on $X$. Since $H^2(X,{\mathbb R})=0$, it follows that it is $d$-exact (in this particular case you can write explicitely the "primitive" of $p^*\omega_{{\mathbb P}^2}$). This shows that $X$ cannot be sG (strongly Gauduchon). Indeed, a result of Popovici states that $X$ is non-sG if and only if $X$ supports a non-zero, closed, positive $(1,1)$-current which is $d$-exact. And here we have $p^*\omega_{{\mathbb P}^2}$, a positive, $d$-exact $(1,1)$-current.