2018/05/31 by Ping Li · 6 citations
Mathematics · #Ball (mathematics) #Bundle #Canonical bundle #Conjecture #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Hyperbolic 3-manifold #Hyperbolic manifold #Manifold (fluid mechanics) #Rigidity (electromagnetism) #math.AG #math.DG #msc:32Q45 #msc:57R20 #msc:58J20
paper · pdf · doi:10.1090/tran/7955
published in Transactions of the American Mathematical Society 372(10), 6853-6868 (American Mathematical Society) · 16 pages, to appear in Transactions of the AMS
openalex created_date 2018/06/01 · arxiv created 2019/07/29 · openalex publication_date 2019/07/31 · arxiv updated 2019/09/10 · openalex updated_date 2026/08/05
We show in this article that Kähler hyperbolic manifolds satisfy a family of optimal Chern number inequalities and that the equality cases can be attained by some compact ball quotients. These present restrictions to complex structures on negatively curved compact Kähler manifolds, thus providing evidence for the rigidity conjecture of S.-T. Yau. The main ingredients in our proof are Gromov’s results on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Hodge numbers, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -phenomenon of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi Subscript y"> <mml:semantics> <mml:msub> <mml:mi> χ </mml:mi> <mml:mi>y</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">χ y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -genus and Hirzebruch’s proportionality principle. Similar methods can be applied to obtain parallel results on Kähler nonelliptic manifolds. In addition to these, we term a condition called “Kähler exactness”, which includes Kähler hyperbolic and nonelliptic manifolds and has been used by B.-L. Chen and X. Yang in their work, and we show that the canonical bundle of a Kähler exact manifold of the general type is ample. Some of its consequences and remarks are discussed as well.