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  Hermite-Einstein metrics with potential

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Let $(M,\omega)$ be a Kaehler manifold, an holomorphic fiber bundle $E$ is Hermite-Einstein with potential $\phi \in \Lambda^1(M)\otimes End(E)$ if there are a hermitian metric $h$ over $E$, and a Chern connection $\nabla$ such that:

$$\Lambda (F(\nabla)+d^{\nabla}\phi)=\lambda Id$$

with $F(\nabla)$, the curvature of the Chern connection and $\Lambda$, the contraction with $\omega$, $\lambda$ is a constant.

Have we an Hermite-Einstein metric for any potential $\phi$?

asked May 27 in Mathematics by Antoine Balan (475 points) [ no revision ]

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