# Is Thierry Aubin’s theorem true on Hermitian manifolds?

A classical theorem of Thierry Aubin states that:

Theorem (Aubin, T. 1979): If the Ricci curvature of a compact Riemannian manifold is non-negative and positive at a point, then the manifold carries a metric of positive Ricci curvature.

In the study of structures on manifolds (such as Hermitian, Kaehlerian, symplectic,...) is the above theorem true? e.g.

Question: Does "the Ricci curvature of a compact Hermitian manifold is non-negative and positive at a point", imply "the manifold carries a Hermitian metric of positive Ricci curvature"?

EDIT: The content below and at the comments rely on Kahler manifolds, and this does not intend to be a complete or even satisfactory answer:

The case when the first Chern class is positive is rather delicate. It was disproved by Yau that even provided the manifold possess positive first Chern class, then there is no ensure for example that the manifold admits a positive Kähler Einstein metric. Also, even when the Kähler–Einstein metric exists, it need not be unique.

Further, a necessary condition for the existence of a Kähler–Einstein metric is that the Lie algebra of holomorphic vector fields is reductive. It was conjectured by Yau that provided the first Chern class is positive, a Kähler manifold has a Kähler–Einstein metric if and only if it is stable in the sense of geometric invariant theory.

More nice comments appear here: First chern class

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A necessary condition (assuming Kähler) to this rely on the fact the first Chern class of manifold is positive. Since the Ricci curvature of an hermitian metric is connected with its Ricci form, that determines the first Chern class.

On the otherside, provided $c_1(M) > 0$ and the manifold is compact and Kähler, then by the Calabi-Yau theorem your requirement is true.

• So does the condition "Ricci is nonnegative everywhere and positive at a point" imply the first Chern class is positive? – Fan Zheng Sep 10 '17 at 2:19
• If I am not mistaken just in the Kahler case, sorry for not pointing it out. – L.F. Cavenaghi Sep 10 '17 at 2:37
• Which Calabi-Yau theorem are you referring to? If I remember correctly, there are additional conditions needed if $c_1(M)>0$, see wikipedia. – Sebastian Goette Sep 10 '17 at 16:54
• @SebastianGoette, If I am not mistaken, provided $c_1(M) > 0$, $M$ is compact and Kaehler, we have a $(1,1)$-form $\rho'$ at the same cohomology class of the Ricci form $\rho$ of the initial K\"ahler metric on $M$. By Calabi-Yau theorem we can find a K\"ahler metric such the Ricci form is $\rho'$. – L.F. Cavenaghi Sep 10 '17 at 17:38
• I think the point of the question is asking what is known assuming only Hermitian and not necessarily Kahler. – Deane Yang Sep 10 '17 at 19:00