ON STROMINGER KAHLER-LIKE MANIFOLDS WITH DEGENERATE TORSION

作者：Yau, Shing-Tung; Zhao, Quanting^*; Zheng, Fangyang^*

来源：Transactions of the American Mathematical Society, 2023, 376(5): 3063-3085.

DOI：10.1090/tran/8659

• Ka
• hler-like
• Strominger connection
• Chern connection
• Riemannian connection
• pluriclosed metric
• locally conformal Ka
• hler
• Sasakian manifold
• degenerate torsion

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Ka.hler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Ka.hler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Ka.hler manifold. Previously, we have shown that any SKL manifold (Mn, g) is always pluriclosed, and when the manifold is compact and g is not Ka.hler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Ka.hler SKL manifolds in dimension 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Ka.hler SKL manifold, its Ka.hler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Ka.hler when n >= 3, and the manifold does not admit any Hermitian symplectic metric.

• 单位
  重庆师范大学