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Positivity and the Kodaira embedding theorem

Lei Ni and Fangyang Zheng

Geometry & Topology 26 (2022) 2491–2505
DOI: 10.2140/gt.2022.26.2491
Abstract

The Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. X Yang (2018) proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms of its curvature. We prove that any compact Kähler manifold with positive 2 nd scalar curvature (which is the average of holomorphic sectional curvature over 2–dimensional subspaces of the tangent space) must be projective. In view of generic 2–tori being nonabelian, this new curvature characterization is sharp in certain sense.

Keywords
Kähler manifolds, projective embedding, compact complex manifolds, Kähler metrics, positive holomorphic sectional curvature, $k^{\text{th}}$ scalar curvature
Mathematical Subject Classification 2010
Primary: 53C55
Secondary: 53C44
References
Publication
Received: 21 January 2020
Revised: 13 April 2021
Accepted: 8 June 2021
Published: 13 December 2022
Proposed: Simon Donaldson
Seconded: Gang Tian, Tobias H Colding
Authors
Lei Ni
Department of Mathematics
University of California, San Diego
La Jolla, CA
United States
Fangyang Zheng
School of Mathematical Sciences
Chongqing Normal University Chongqing
China