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Quasi-Negative Holomorphic Sectional Curvature and Ampleness of the Canonical Class

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Complex and Symplectic Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 21))

Abstract

This note is an extended version of a 50 min talk given at the INdAM Meeting “Complex and Symplectic Geometry”, held in Cortona from June 12th to June 18th, 2016. What follows was the abstract of our talk.

Let X be a compact Kähler manifold with a Kähler metric whose holomorphic sectional curvature is strictly negative. Very recent results by Wu–Yau and Tosatti–Yang confirmed an old conjecture by S.-T. Yau which claimed that under this curvature assumption X should be projective and canonically polarized. We will explain how one can relax the assumption on the holomorphic sectional curvature to the weakest possible, i.e. non positive and strictly negative in at least one point, in order to have the same conclusions. We shall also try to motivate this generalization by arguments coming from birational geometry, such as the abundance conjecture.

The results presented here were originally contained in the joint work with Diverio and Trapani (Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle, 2016, ArXiv e-prints 1606.01381v3).

In memory of Paolo De Bartolomeis

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Acknowledgements

We would like to warmly thank the organizers Daniele Angella, Costantino Medori, Adriano Tomassini for the beautiful and stimulating environment of the INdAM Meeting “Complex and Symplectic Geometry”, in Cortona. A particular thought goes to Paolo De Bartolomeis, who sadly passed away on November 29th, 2016.

The author is partially supported by the ANR project “GRACK”, ANR-15-CE40-0003, and the ANR project “Foliage”, ANR-16-CE40-0008.

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Authors and Affiliations

  1. Dipartimento di Matematica “Guido Castelnuovo”, SAPIENZA Università di Roma, Piazzale Aldo Moro 5, I-00185, Roma, Italy

    Simone Diverio

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  1. Simone Diverio
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Correspondence to Simone Diverio .

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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica ‘Ulisse Dini’, University of Florence, Firenze, Italy

    Daniele Angella

  2. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, University of Parma, Parma, Italy

    Costantino Medori

  3. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, University of Parma, Parma, Italy

    Adriano Tomassini

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Diverio, S. (2017). Quasi-Negative Holomorphic Sectional Curvature and Ampleness of the Canonical Class. In: Angella, D., Medori, C., Tomassini, A. (eds) Complex and Symplectic Geometry. Springer INdAM Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-62914-8_5

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