Let Ω be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new "normal families" arguments in the construction of the limit biholomorphism.

Abstract. We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.

Abstract. In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.

It is a basic tenet in complex geometry that negative curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while positive curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifold M with a Kähler metric with positive total scalar curvature is uniruled, which is equivalent to every point of M being contained in a rational curve. We also prove that if M possesses a Kähler metric of total scalar curvature equal to zero, then either M is uniruled or its canonical line bundle is torsion. The proof of the latter theorem is partially based on the observation that if M is not uniruled, then the total scalar curvatures of all Kähler metrics on M must have the same sign, which is either zero or negative.

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