# Projectively flat Hermitian curvature proportional to Kähler form?

Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to the Kähler 2-form: $$R^Ee=s\,\omega\otimes e \ ,\ \ \forall e\in E\ .$$

Projectively flat means the curvature 2-form of the Chern connection on $E$ is a multiple $\alpha 1_E$ of the identity of $E$ by a $(1,1)$-form $\alpha$ (see e.g. famous book S.Kobayashi, Differential Geometry of Complex Vector Bundles, Math. Soc. of Japan, Iwanami Schoten and Princeton UP, 1987). But what about when $\alpha=s\omega$, a scalar multiple of the Kähler 2-form not necessarily closed? And what about if it is closed?

If $E=TM$ and $M$ is Kähler, and, moreover, $s$ is constant, then the answer is the constant holomorphic sectional curvature manifolds.

Thank you very much for answers.

• You are mentioning Proposition 1.2.8 of mathsoc.jp/publication/PublMSJ/PDF/Vol15.pdf . When you have holomorphic surjective map $\pi:X\to \Delta$, and fibers admit $c_1(X_s)<0$, then you are facing with variation of Kahler-Einstein metrics, namely, $Ric(\omega_s)=-\lambda_s\omega_s$, where $\lambda_s$ is fiberwise constant(which may not be constant on the whole of $X$), hence in this case relative tangent sheaf $T_{X/\Delta}$ if being projectively flat then we always have $Ric_{X/\Delta}\omega=\lambda\omega$, where you can take the ristriction of $\lambda$ on each fiber be $-1$. – user21574 Oct 28 '17 at 13:32
• In previous comment I used Proposition 4.1.11 of mathsoc.jp/publication/PublMSJ/PDF/Vol15.pdf – user21574 Oct 28 '17 at 13:39
• When fibers are hyperbolic Riemann surface, then relative tangent sheaf is projectively flat iff $Ric_{X/\Delta}\omega=-\lambda\omega$ on holomorphic surjective map $\pi:X\to \Delta$ – user21574 Oct 28 '17 at 13:53

The curvature form of any bundle is closed by Bianchi identity, hence your form $s\omega$ is necessarily closed. This implies that $s=const$, unless $dim_C M=1$, because $ds\wedge \omega=0$ implies $ds=0$ (multiplication by $\omega$ is injective on 1-forms).
As for the main question, projectively flat bundles are the same as flat PGL(n)-bundles, which is the same as flat bundles with ${\Bbb P}^n$-fibers. They are classified by homomorphisms from $\pi_1(M)$ to $PGL(n)$ up to $PGL(n)$-action.
• Dear Misha Verbitsky, in my mind there was only one question and "Kähler form" is just the associated 2-form $\omega$ on Hermitian manifolds, not necessarily closed. Thanks for the answer. Regarding the classification you refer, the subclass of connections I am looking for is strictly smaller, I guess. And still interesting... – Albuquerque Jul 12 '14 at 16:18