I'm studying $4$-dimensional manifold, we know that the hodge star operator give the poincare duality. And in dimension $4$ it gives the intersection form.
My question is, does there exist a smooth oriented four dimensional manifold $X$, for any cohomology class $\alpha\in H^2_{hodge}(X,\mathbb{R})$, we can find a metric $g_{\alpha}$ of $X$ such that $*_{g_{\alpha}}\alpha=-\alpha $ ? i.e. every cohomology class can be in the negative eigenvector space for some hogde star operator.
What's more. Does there exist a manifold and two metric $g_1$ and $g_2$ such that $\alpha$ is eigenvector with eigenvalue $1$ for $g_1$ but it is eigenvector with eigen value $-1$ for $g_2$?
I believe this is not true but I don't find a way to disprove it.
Thank you for your answer.
I think the second question is more obvious since
$0<\int_M\alpha \wedge *_{g_1}\alpha=\int_M \alpha\wedge \alpha$
and
$0<\int_M\alpha\wedge *_{g_2}\alpha=-\int_M\alpha \wedge \alpha$
Which leads a contradiction.