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On the coherence conjecture of Pappas and Rapoport

We prove the (generalized) coherence conjecture of Pappas and Rapoport. As a corollary, one theorem of Pappas an Rapoport, which describes the geometry of the special fibers of the local models for ramified unitary groups, holds unconditionally. Our proof is based on the study of the geometry (in particular certain line bundles and $\ell$-adic sheaves) of the global Schubert varieties, which are the (generalized) equal characteristic counterparts of the local models.