Positivity and the Kodaira embedding theorem

Ni, Lei¹ ; Zheng, Fangyang²

¹ Department of Mathematics, University of California, San Diego, La Jolla, CA, United States
² School of Mathematical Sciences, Chongqing Normal University Chongqing, China

Geometry & topology, Tome 26 (2022) no. 6, pp. 2491-2505.

Résumé

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 $2^{\text{ nd}}$ 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.

Détail

Classification : 53C55, 53C44
Keywords: Kähler manifolds, projective embedding, compact complex manifolds, Kähler metrics, positive holomorphic sectional curvature, $k^{\text{th}}$ scalar curvature

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