论文标题
在有限体积双曲线歧管的贝蒂数量上
On the Betti Numbers of Finite Volume Hyperbolic Manifolds
论文作者
论文摘要
我们获得了紧凑的复合氧化歧管的贝蒂数量的强上限。我们使用统一的固体来改善[DS17]技术最直接应用所给出的结果。我们还为大多数程度的紧凑型Quaternionic-和Cayley-Hyperbolic歧管提供有效的上限。更重要的是，我们将技术扩展到完成有限体积的实量和复杂的氧化折射。在这种情况下，我们为双曲线尖端上强烈的谐波形式产生了新的单调性不平等，并采用了新的峰值论点来估计$ l^2 $ - 生物学等级。最后，我们利用$ l^2 $ colemology的边界的结合，在体积上的cusps数量以及对某些等级的$ l^2 $ - 酒精学对某些等级的一本局部对称空间的拓扑解释。
We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective upper bounds for Betti numbers of compact quaternionic- and Cayley-hyperbolic manifolds in most degrees. More importantly, we extend our techniques to complete finite volume real- and complex-hyperbolic manifolds. In this setting, we develop new monotonicity inequalities for strongly harmonic forms on hyperbolic cusps and employ a new peaking argument to estimate $L^2$-cohomology ranks. Finally, we provide bounds on the de Rham cohomology of such spaces, using a combination of our bounds on $L^2$-cohomology, bounds on the number of cusps in terms of the volume, and the topological interpretation of reduced $L^2$-cohomology on certain rank one locally symmetric spaces.
