Journal of Differential Geometry

Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle

Simone Diverio and Stefano Trapani

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We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non-positive and strictly negative in at least one point, then its canonical bundle is positive. This answers in the affirmative to a question first asked by S.-T. Yau.


The first-named author is partially supported by the ANR Programme: Défi de tous les savoirs (DS10) 2015, “GRACK”, Project ID: ANR-15-CE40-0003ANR, and Défi de tous les savoirs (DS10) 2016, “FOLIAGE”, Project ID: ANR-16-CE40-0008.

Article information

J. Differential Geom., Volume 111, Number 2 (2019), 303-314.

Received: 4 August 2016
First available in Project Euclid: 6 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q15: Kähler manifolds
Secondary: 32Q05: Negative curvature manifolds

holomorphic sectional curvature Monge–Ampère equation canonical bundle


Diverio, Simone; Trapani, Stefano. Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle. J. Differential Geom. 111 (2019), no. 2, 303--314. doi:10.4310/jdg/1549422103.

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