Ricci Curvatures on Hermitian manifolds

By Kefeng Liu, Xiaokui Yang. 2014

96 citations recorded by the source corpus.

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $S^{2n-1}\times S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifolds such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1aa2bda13ac8f58626afc21184f8d8e6a71f8eca

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In Applied Mathematics

  1. Cristiano Spotti
  2. Simone Calamai
  3. Daniele Angella
  4. J. Stelzig

In Astronomy and Astrophysics

  1. Cristiano Spotti
  2. Daniele Angella

In Geometry and Topology

  1. Cristiano Spotti
  2. Simone Calamai
  3. Daniele Angella
  4. J. Stelzig
  5. Y. Odaka

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