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Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

Frédéric Campana;  Henri Guenancia;  Mihai Păun — 2013

Annales scientifiques de l'École Normale Supérieure

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.

Non-deformability of entire curves in projective hypersurfaces of high degree

Olivier Debarre;  Gianluca Pacienza;  Mihai Păun — 2006

Annales de l’institut Fourier

In this article, we prove that there does not exist a family of maximal rank of entire curves in the universal family of hypersurfaces of degree d ≥ 2 n in the complex projective space ℙ n . This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.

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