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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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Closed parallel (1,1)-forms on compact Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold. We know one example of a closed parallel (1,1)-form, namely, $\omega$ itself. Are there obstructions for the existence of non-vanishing closed parallel ( …