Kählerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures

TL;DR

This work shows that on compact Hermitian surfaces, semi-definite Strominger-Bismut-Ricci curvatures force Kählerness under precise conditions. It develops new Ricci-curvature and Chern-number identities for the Strominger-Bismut connection and then applies them to derive, under various sign and Gauduchon-parallel torsion scenarios, that the Hermitian metric must be Kähler. In stronger cases, the results further classify the surface as projective or Calabi–Yau. The findings extend Yang's results and provide boundedness-based criteria that guarantee Kählerness from SB-curvature data.

Abstract

We prove that a compact Hermitian surface is Kähler under certain non-positivity or non-negativity conditions on Strominger-Bismut-Ricci curvatures. The key tools for achieve these results are new Ricci curvature and Chern number identities for the Strominger-Bismut connection. This work complements and extends earlier results of Yang.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1: see, e.g., Lemma 3.4 in LY17
• Lemma 2.2: Yang2507
• Remark 2.3
• Lemma 2.4: WY25Yang2508