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In this note we show that if a projective manifold admits a Khler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

It is known that if the special automorphism group SAut(X) of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group Aut(X) to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial Ga or Gm-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.

In this paper, we study the <i></i>-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in [12], we show that <i></i>-ordinary locus is a union of certain maximal EkedahlKottwitzOortRapoport strata introduced in [12] and we give criteria on the density of the <i></i>-ordinary locus.