学术文献库

This survey article mainly addresses to graduate students and young researchers in complex geometry willing to enter the beautiful word of connections between curvature and Kobayashi hyperbolicity. It is a detailed account of a recent breakthrough by Wu and Yau, shortly after generalized by Tosatti and Yang (and others), which sits on the crossroad between complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture by Kobayashi, stated at the very beginning of the theory, predicts that a compact hyperbolic manifold should have ample canonical bundle. Now, on the one hand it is also known since the beginning of the theory that a compact complex manifold with a Hermitian metric whose holomorphic sectional curvature is negative is Kobayashi hyperbolic. On the other hand a compact Kähler manifold with ample canonical bundle is known -- by the celebrated work of Aubin and Yau -- to admit a Kähler metric with (constant) negative Ricci curvature. Wu and Yau's theorem states that if a smooth projective manifolds admits a Kähler metric with negative holomorphic sectional curvature, then it also admits a possibly different Kähler metric whose Ricci curvature is negative. It can be therefore seen as a weak confirmation of Kobayashi's conjecture above, since it gives the same conclusion but with the stronger hypothesis about the holomorphic sectional curvature. Beside a fully detailed presentation of the proof of this theorem, we also provide some basic background on complex differential geometry as well as several (positive or negative) results about the theme of curvature and hyperbolicity. Some natural open questions are also discussed. The proof of the Wu-Yau theorem presented here follows quite closely the original main key ideas by Wu and Yau, but the conclusion is somehow simplified using the pluripotential approach of the author and S. Trapani.