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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 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
Primary: 53C55
Secondary: 53C44
##### 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