Let $(M,\omega)$ be a compact Kähler manifold with Kähler form $\omega$. Furthermore, denote by $c\_{1}$ the first Chern class of $M$. Assume one of the following $c\_{1}>0$, $c\_{1}<0$ or $c\_{1}=0$. My question is the following.


**Question:** How can one show that there exists a real number $\lambda \in \mathbb{R}$ and a Kähler form $\tilde{\omega}$ such that $\lambda \tilde{\omega} \in c\_{1}$?


Greetings,
Ben



First Chern class with sign
See Demailly, **Complex Algebraic and Differential Geometry**, p.333. By definition, the first Chern class of a vector bundle is positive (negative, zero) if it is positive (negative, zero) as a cohomology class, i.e. representable by some positive (negative, zero) $(1,1)$-form, i.e. a closed $(1,1)$-form which is positive (negative, zero) on any complex line in the tangent space. If positive (negative), we can take $\lambda=1$ ($\lambda=-1$) and that form as $\tilde\omega$; the manifold $M$ need only be a compact complex manifold, not necessarily Kaehler. If zero, we can take $\lambda=0$ and any Kaehler form as $\tilde\omega$, but we must then assume that $M$ is Kaehler. 


