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Aug 8 at 8:29 comment added abx Well, you can say that $\tau +d\omega $ is positive for some real smooth 1-form $\omega $. I don't see what else you can hope for.
Aug 8 at 8:21 comment added Tom @abx, we can say $\tau+i\partial\bar\partial f$ is positive for some real smooth function $f$ on $X$, but for the non-Kähler case, we can't find such an $f$.
Aug 8 at 5:54 comment added abx Suppose that $X$ is Kähler, and $[T]$ is a Kähler class. What can you say from $\tau $, apart that it is cohomologous to a Kähler form?
Aug 8 at 4:03 history asked Tom CC BY-SA 4.0