Scalar curvature on compact complex manifolds

Abstract

In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if $K_X$ (resp. $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\ge 2$, there exist smooth Hermitian metrics with positive total scalar curvature and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti and B. Weinkove.

To appear in Tran. Amer. Math. Soc

Cited by in corpus (4)

• RC-positivity, rational connectedness and Yau's conjecture
• Levi-Civita Ricci-flat metrics on compact complex manifolds
• Minimal complex surfaces with Levi-Civita Ricci-flat metrics
• RC-positivity and scalar-flat metrics on ruled manifolds