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.

  • 1
    $\begingroup$ 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 $-1$. $\endgroup$ – user21574 Oct 28 '17 at 13:32
  • 1
    $\begingroup$ In previous comment I used Proposition 4.1.11 of mathsoc.jp/publication/PublMSJ/PDF/Vol15.pdf $\endgroup$ – user21574 Oct 28 '17 at 13:39
  • 1
    $\begingroup$ When fibers are hyperbolic Riemann surface, then relative tangent sheaf is projectively flat iff $Ric_{X/\Delta}\omega=-\lambda\omega$ on holomorphic surjective map $\pi:X\to \Delta$ $\endgroup$ – 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.

  • $\begingroup$ 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... $\endgroup$ – Albuquerque Jul 12 '14 at 16:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.